Q mmMmmm#m* ^>o..^S^.^.vv.>Z/\V.N/>vn so\N Vs\x Va\ v5o\n -*/X< ~ K/^A&^'y\V,(^\£<^^ 1 vol. evo. ^^ ^*C'V>Kv-a^@^>^i u A complete manual for the plank and rail road builder." ^#1^^^- > MANSFIELD (Edward D.) ON AMERICAN EDUCATION. i'^^^^^W ^■^Mm^^^^ 1 vol. 1-'.no. ^^^x^^^Sk^: va^v'X'^C^ ? '^^F'^^'^^i^d'^^' ICE OF TEACHING, ^^^^m^^m r^A^?^A^ ^.Vs- -° v worK OI i;'vai practical vaiue to every miei M^^^iMfe^S PAGE'S (David P.) THEORY AND PRACT] :A/^i:Y^^S^^hs^h'ic * Vo'- '-'no. vi0 It is a grand book ; and I thank Heaven that you have written it." ^-%^:^d' :^ ^Wp^§MW^ DA VIES' (Charles, LLP.) LOGIC AND UTILITY OF M AT IIE-JM||§g|i||M HvmWt^m* maitcs. ivoi.hvo. ^^w^-mw^ ..fyr.y,st ^j&^<&£■Y>h^ A valuable book for every student's and teacher's library. r^--^Si--PW^:^0^ri^i:-y^r ^^^:;^lii$ COLTOX (Rev. Calvin) OX PUBLIC ECOXOMY FOR THE W^^^^MM %^%Z^mM^ UNITED STATUS. 1 vol. 8^. t^^M^//^r^Sb^^-P *" Every free-trade man should not fail to read it." very p;i^ ful aphorisms—a book that will never be out of date." ^ffMMI|TCOLT()X'S(Rev. Walter) THREE YEARS IX CALIFORNIA. ^mMm'MWt ?fe^Sllx^^4> [ vul- J~mo- Wi,h ina,,s and en"ravm-s- &&&cM-AA^h&^4 All interested in this new country will perusr it with delight. ^ #■ ^K^IKMAXSFIELD'S (Ed. D.) HISTORY OF THE MEXICAN WAR t§^k^xW:^> 1 vol. l-Jmo. With maps and engraving. M'W^Wi " The nh,st uUth('Mtir work I>"1>1^ll<,,l >" tins country." ^■p^.^^:r^:^^ ^•MiS^I^7^^ MANSFIELD'S (Edward D.) LIFE OF GEN. WINFJKLD SCOTT. tl4^^S#|^ ««5P I vol. l-2mo. With illustrations. ^fe^^^^^ mmmmm* WILLARD'S (Mrs. Emma) HISTORY OK THE U. STATES. *Mf^|^^VS>. i^^M^M^ WILLARDS(Mrs. Enuna) UNIVERSAL HISTORY in Perfective. ^^^-^30^ ftl?ifif^l GOULD'S (Edward S.) ABKIJXJMENT OF ALISOX'S EUROPE. W^MiMmM ilHS^M^? PARKEIl'S (Ric-hard'G.) RHETORICAL READER, i-mo. ^m^-^'^^Mi iii?^;^^X KIX(;SLEYS (George) CHURCH . ^^^^^i?(-I,ARK-S (S. W.) NEW ENGLISH GRAMMAR ^^W^^aM^^ :§£ m^^W^'-- BROOKS' (Nj.tlian t1.) GREEK AND LATIN CLASSICS. *^Mt-M^-&M<& VXD SCHOOL MUSIC BOOKS. J|f.^^M^S:fef. l'imo. ^M^^ili^OIlTHEXD'S (Charles) ELOCUTIONARY WORKS. ^ vols. %MM^^^^^ '^0MW^^ McIN'ITRE'S (James, M. D.) TREATISE ON ASTRONOMY AND ^WM^^&^M "^^^m^Ma^ USE OF THE GLOBES. fMMMWMM • I BARTLETT'S ACOUSTICS AND OPTICS. 3 ELEMENTS OF NATURAL PHILOSOPHY. BY W. H. C. BARTLETT, LL. D., PROFESSOR OF NATURAL AND EXPERIMENTAL PHILOSOPHY IN THE UNITED STATES MILITARY ACADEMY AT WEST POINT. II.-ACOU STIC S. II I.-OPTIC S. NEW YORK: PUBLISHED BY A. S. BARNES AND COMPANY, No. 51 JOHN-STREET. CINCINNATI: H. W. DERBY AND COMPANY. 18 52. Entered according to Act of Congress, in the year One Thousand Eight Hundred and Fifty-tvvo, By W. II. C. BARTLETT, In the Clerk's OfRce of the District Court of the United States for the Soutliern District of New-York. J. P. JONZS it CO., STEREOTTPERS, 183 AVilliam-street. G. W. WOOD, PRINTER, 51 Jolm-st, cor. Dutch. PREFACE. Those who are familiar with the subjects of which the present volume professes to treat, will readily recognize the ^*ces whence most of its materials are drawn. In the use ase materials, no distinction of principle is made between . jnd and light. Both are regarded and treated as the effects of certain disturbances of that particular state of mole- cular equilibrium which determines the ordinary condition of natural bodies; the only difference being in the media through which these disturbances are propagated, and in the organs of sense by which their effects are conveyed to the mind. The study of Acoustics is, therefore, deemed to be not only a useful, but almost a necessary preliminary to that of Optics. In the preparation of the part relating to Sound, great use was made of the admirable monograph of Sir John HePwSCHel, published in the Encyclopaedia Metropolitana; and whenever it could be done consistently with the plan of the work, no hesitation was felt in employing the very language of that 4 PREFACE. eminent philosopher. Much valuable matter was also drawn from Mr. Airy's Tracts, and from the labors of Mr. Kobison and M. Peschel. In addition to the works of the authors just cited, those of Mr. Coddington, Mr. Powell, Mr. Lloyd, Sir David Brew- ster and M. Babinet were freely consulted in constructing the part relating to Optics. TABLE OF CONTENTS, ACOUSTICS. Introductory Remarks and Definitions, Waves in general, .... Velocity of Sound in Aeriform Bodies, Velocity of Sound in Liquids, . • Velocity of Sound in Solids, Pitch, Intensity, and Quality of Sound, Siren, ..... Divergence and Decay of Sound, Molecular Displacement, Interference of Sound, New Divergence and Inflexion of Sound, . Reflexion and Refraction of Sound—Echos, Hearing Trumpet, Whispering Galleries and Speaking Tubes, Musical Sounds, .... Vibrations of Musical Strings, Monochord, .... Vibrating Columns of Air, . Vibrations of Elastic Bars, Vibrations of Elastic Plates and Bells, Communication of Vibrations, The Ear, ...••• Music, Chords, Intervals, Harmony, Scale, and Temperament, Table of Intervals, ..... Table of Intervals with the Logarithms, . 6 TABLE OF CONTENTS OPTICS. Introductory Remarks and Definitions, Reflexion and Refraction of Light, Table of Refractive Indices and Refractive Powers, Deviation of Light at Plane Surfaces, Deviation of Light at Spherical Surfaces, Deviation of Light by Spherical Lenses, Deviation of Light by Spherical Reflectors, Spherical. Aberration, Caustics and Astigmatism, Oblique Pencil through the Optical Centre, Optical Images, .... The Eye and Vision, .... Microscopes and Telescopes, Common Astronomical Telescope, . • Galilean Telescope, Field of View, .... Terestrial Telescope, Compound Refracting Microscope, Astronomical Reflecting Telescope, Gregorian Telescope, . • Cassegrainian Telescope, . Newtonian Telescope, Dynameter, .... Camera Lucida, .... Camera Obscura, • . . Magic Lantern, .... Solar Microscope, Chromatics, ..... Color by Interference, Color of Gratings, . Table of Wave Lengths, Colored Fringes of Shadows and Apertures, Colors of Thin Plates, Colors of Inclined Glass Plates, Colors of Thick Plates, Tage 1G7 171 176 176 188 201 211 216 220 222 232 238 245 246 248 250 252 253 254 256 257 257 261 263 264 265 266 267 276 282 284 290 291 TABLE OF CONTENTS. 7 Page Color from Unequal Ref tangibility, . . . . 293 Dispersion of Light, . . . . . . .298 Table of Dispersive Powers, . . . . . 301 Chromatic Aberration, . ..... 302 Achromatism, . . . . . . • 305 Internal Reflexion, . . , . . . .309 Absorption of Light, . . . . . . #11 The Rainbow, . . . . • . .314 Halos, . . . . • • • • 321 Polarization of Light, . . . . . .322 Polarization by Reflexion and Refraction, . . . . 327 Polarization by Absorption, . . . . . .334 Double Refraction, . . . . . • 335 Circular Polarization, . . . . • • .345 Chromatics of Polarized Light, . • . • • 348 ELEMENTS OF ACOUSTICS. acous- §1. The principle which connects us with the external world through the sense of hearing, is called sound ; and sound. that branch of Natural Philosophy which treats of sound, -n -i A Acoustics. is called Acoustics. To explain,4;he nature of sound, the laws of its propaga- tion through the various media which convey it to our ears, the mode of its action upon these organs, the modifi- objects of cations of which sound is susceptible in speech, in music tics- and in unmeaning noise, as well as the means of pro- ducing and regulating these modifications, are the objects of acoustics. 8 2. All impressions derived through the senses, imme-0rigin of a11 our ox o 5 impressions, diatelv follow and may, therefore, be said to arise from peculiar conditions of relative motion among the elements of which certain parts of our physical organization are constructed. These conditions are mainly determined by Conditions t0 t t /» i -i t 'ii.i • " cause sensation the internal state ot the bodies with which we are in sen- determined. sible contact. It becomes important, therefore, to look somewhat into the constitution of natural bodies, and especially to inquire into the structure of the atmosphere, the main vehicle of sound to the beings which inhabit it, vehicle of sound; and of that vastly more subtile and refined medium, called ether, through which we derive our sensations of heat, light, and electricity. 0f usht» &c- § 3. We have already referred, in the introduction to the first volume, to Boscovich's views upon this subject, and shall now give some illustration of the mode in which, 10 NATURAL PHILOSOPHY. Exponential curve; according to that distinguished philosopher, all bodies are formed. For this purpose let us resume the exponential curve as exhibited in the annexed figure, and which Boscovich sup- Fig. 1. Attractive ordinates; Repulsive ordinates; j) ft poses to represent the law and intensity of the action of one atom of a body upon another. We have.seen that the ordinates of those portions of the curve which lie above the line A C, denote the attractive, while the ordinates of the portions below, represent the repulsive energies of an atom A for another atom situated anywhere upon this line. That at the points C, Df, C", D", in which the curve intersects the line A C, the reciprocal action of the atoms reduces to nothing, and the atoms become neutral. Also that an atom situated at Dr, D," or D,"f and the atom A Neutral points; constitute a temporary molecule, while the molecule formed of the atoms A and C, A and C", or A and Cf", has a cer- tain degree of permanence, resisting compression and dila- tation, and tending to regain its original bulk when the distending or compressing cause is withdrawn. But this permanence only obtains when the disturbing force is such as to change the interval between the atoms by a distance less than that which separates the consecutive positions of neutrality ; for if the molecule A C", for example, be com- pressed into a less room than A Df, the atom originally at C", will not return to that point, but will be attracted by A, and the molecule will tend to collapse into the bulk A C. HA C" be stretched beyond the bulk A D", it will tend to take the dimension A C". The only mole- cule that cannot be permanently changed bv compression is A C. Temporary molecule; Permanent molecule ; 'When permanence exists. ELEMENTS OF ACOUSTICS. 11 Fig. 2. F' -U' lyW Fig. 3. The component atoms of molecules thus constituted are, when in a state of relative equilibrium, in a condition of inactivity upon each other. The approximation or sepa- How the redpro- ration of the atoms by the application of some extraneouscalactlonsamons */ xx atoms are ex- cause, gives rise to the exertion of the repulsive or at-cited. tractive forces inherent in the atoms, and thus these forces may be said to be excited or brought into action. The compression or dilatation is the occasion, not the efficient cause of the attractions and repulsions among the atoms. § 4. The intensity of the atomical forces determines the Form of the ex- &/» , i . • i ti? ponential curve rm ot the exponential curve. 11 a determined. very moderate force produce a sensi- ble displacement of the atoms, the ordinates E' dr, and Ed, on each side of the position C, of inactivity, must be short, and the exponential curve will cross the axis very obliquely, in order that the ordi- nates expressing the attractive and repulsive forces may increase slowly. If, however, it require great force to produce a sensible compression or dilatation, the curve must cross the axis almost perpendicularly. But in every case it must be remarked, and the remark is most small compres- important, that when the compression or distension bears6 a small proportion to the distance between the neutral positions of the atoms, the degree of compression or dis- tension will be sensibly propor- tional to the intensity of the dis- turbing force. For, when the displacement D' E or D' E' is very small in comparison to C D', the elementary arc dD'd' will sensibly coincide with a straight line, and the ordinates E d and Ef df, be proportional to the compression Df E or distension Df E'. That is to say, because action and sion. A A \ yyr js ^\ S' d^~ 12 NATURAL PHILOSOPHY. Fig. 4 Their reaction are equal, a disturbed consequences. afom WiU be Urged back to- wards its position of neutrality by a force whose intensity is proportional to the distance of the atom from that point. Moreover, supposing the atom A, Fig. 4, to be kept station- ary, and the points E, and E\ to mark the limits of the disturbance of the other atom, this latter will return to its position of neutrality Df, with a living force due to the action of the force of restitution over the path ED', or E' D'; it will, therefore, pass the point D\ after which the direction of the action will be reversed, the living force will be destroyed, the atom will again return to its Perpetual osciiia-Position °f neutrality, which it will pass as before, and for won; the same reason, and thus be kept in perpetual oscillation. But the action between the two atoms of the molecule be- ing reciprocal, the atom A will not remain stationary, but will move in the same direction as the disturbed atom and tend to preserve its neutral distance, and the oscillation checked. that would otherwise continue will, therefore, be checked. Action of the sim- § 5. Let us next take the case of a molecule of the sim- plest molecule on plest constitution, to wit, one composed of twTo atoms, and examine its action on a third atom situated on the prolon- lation of X Y, joining its elements. Fig. 5. H d' djj First case; Suppose a molecule X Y, composed of the two atoms Xand Y, which are placed, the former at A, and the lat- ELEMENTS OF ACOUSTICS. 13 ter at the last limit of cohesion 6", Fie. 5. The dotted and ExP°neDtial ° curves of the waving curve beginning at Y and running towards G, will component represent the exponential curve of the atom X, in thatatoms; direction, while the similar curve beginning at the point E, will represent that of the atom Y, and the full curve C'A'D'R'C'A" &c, of which the ordinate corres- ponding to any point of the line A C, is equal to the alge- braic sum of the ordinates of the dotted curves correspond- ing to the same point, will be the exponential curve of the That of the molecule X Y, and will give the action of the molecule molecule; upon a third atom placed any where on the line A C be- yond Y. The curve has been carefully constructed ac- cording to the conditions of the case, and shows by simple inspection how different the action of even the simplest molecule is from that of a single atom. The neutral posi- Neutral positions tions of an atom with respect to this molecule will be at° an*on\^lt x respect to this A, O, D', C", D" and so on to G. A curve having a molecule; cusp at A, the middle point of the distance X Y, and diverging so as to be asymptotic with the lines c b and d b',' will give the law and intensity of the action on an atom situated between Xand Y. § 6. If instead of placing the atoms at a distance apart second case; equal to that of the last limit of cohesion from A, as in the last case, we had supposed them separated by the distance A Crt, Fig. 1, the resulting exponential curve would have been still more unlike that of a single atom; for in that case /' Ffc. 6. i\ / •* ■•«-■'' several of the attractive branches, Fig. 6, of one of the atomi- cal curves would have stood opposed to the repulsive Resultingaction % ^ x x on an atom. branches of the other, and the molecule thus rendered in- 14 NATURAL PHILOSOPHY. active on a third atom till the latter be removed nearly to the furthest limit of the scale of corpuscular action. This third atom will, therefore, admit of considerable latitude of displace- ment without much opposition Exemplification, or any great effort to regain its primitive position; a fact we often see exemplified in the class of liquid bodies. Fig. T. Third case; W M B C R' Construction; Construction of the exponential curve giving the action of a molecule on an atom. § 7. Let us now take the molecule composed of two atoms placed at the limits A and C", Fig. 1, and examine its action on a third atom somewhere on the line B B', which bisects at right angles the distance A C". Suppose the third atom placed at z. Join z with A and C", and construct the single atomical curves of A and 0" in reference to z, and suppose the atom z in Fig. 7, to have a position with re- spect to A and On, correspond- ing to any position between D" and C"f, Fig. 1; thus situated, it will be repelled both by A and C'f, Fig. 7. In a pair of dividers take the ordinate z m, Fig. 1, and lay it off from z, on the prolong- ations of Az and Ch'z, Fig. 7, and construct the parallelo- gram z m n m'; the diagonal z n, will represent in direction and intensity the action of the molecule A C" on the third atom. Draw* a perpendic- JR" A!' ELEMENTS OF ACOUSTICS. 15 ular to B Br through the point z, and take the distance z R" equal to z n, the point R" will be one point of the exponential curve of the molecule A 0" in the direction B B\ Other points being determined in the same way, the waved lines of Fig. 7 will indicate the action sought; the ordinates of the branches Af,A",A,n, &c, on one side of B B', denoting attractions, while those of the branches Rr, R", R"', &c, on the opposite side, denote repulsions. We see that this action differs remarkably from that of Action differs a single atom. The curve has, to be sure, like that of a8!68*7. omt at & ' " of an atom. single atom, many alternations of attractions and repul- sions, but these alternations become less marked as they . approach the molecule; and instead of insuperable repul- sion at the greatest vicinity I, we find there a neutral point. Moreover, in approaching the molecule, the repul- sive action ceases at D', where attraction begins and con- tinues, so far as there is any action, all the way through to D! on the opposite side of A C". This molecule is ever active when approached along the line B Bf, except at certain neutral positions where the direction of the action is reversed, and: is easily penetrable in this direction, whereas along the line A C" it exerts little or no action within certain limits, and is capable of an infinite repul- sion within its last limit of cohesion. Thus we see that even in this simplest constitution of a molecule, the action on an atom is susceptible of great variety by mere diffe- rence of position and distance between its component atoms; and it would be easy to show that while the law of the atomic action in all bodies is the same, the reci- _ . 4 . 1 Law of atomic procal action of the molecules com- pounded of these atoms may be un- Fig. a speakably various according to the 7l relative position and distance of the ^ component atoms. § 8. Confining, for the present, the motion of the third atom to the plane of the lines A 0" and B B', action the same in all bodies. Reciprocal action of molecules infinitely various. 16 NATURAL PHILOSOPHY. Fifir. 7, we see that when it is at z, Action of a o " ' molecule on an it is repelled by the molecule A C" \ Fig. 8. atom. Constitution of an elementary surface. when at z' it is attracted, and the action is reduced to nothing at the point D". "When the atom is drawn aside from its neutral position D", say to z", Fig. 8, it will be re- pelled by C" and attracted by A, because the distance from the former will be diminished, while that from A will be increased. Take z" h to represent the intensity of the repulsion and z" o that of the attraction; complete the parallelogram o z" h q, and we shall find the molecule urged to its neu- tral position D" by a force whose intensity and direction are represented by the diagonal z" q; so that, so far as the action in the plane AC" D" is concerned, D" is a posi- tion of stable equilibrium, and the three atoms A, G" and D" will constitute for moderate displacements a permanent molecule, presenting an elementary surface having length and breadth. The same would be true were the third atom placed at Df or D'n, &c, Fig. 7. oscillation of the The disturbed atom when at z" being urged back to its place of neutrality by the molecule A C", will reach that point with a certain amount of living force, due to the ac- tion of the force of restitution over the path from z" to D"; it will, therefore, pass to the opposite side of Dn, where the action being in the opposite direction, its living force will be destroyed, after which it will be brought back and made to oscillate about D" as long as A and Gn are sta- tionary. But while the third atom is on the side z", that at G" will be repelled by it, and that at A attracted; the contrary will be the case when the atom is on the oppo- site side from z", so that the atoms of the molecule A C" will also oscillate, and obviously in such manner as to to cause the neutral position to follow the displaced atom. That of the atoms of the molecule. Explanation of figure; 9. Now conceive a triangle A GT G/\ each of whose ELEMENTS OF ACOUSTICS. 17 sides is equal to a distance at which two atoms may form a permanent molecule, and suppose an atom to be placed at each vertex; these atoms will form a permanent molecule. Place a fourth atom at the vertex D", of a pyramid of which the base is the elementary plane formed by the first three atoms, and each of the edges about the vertex is equal to a distance necessary for two atoms to form a permanent molecule. It will be obvious, from what has already been said, that the fourth atom or that at the vertex cannot be disturbed without being resisted Permanent and urged back to its neutral place by the action of the molecule of four ... . atoms; molecules which form the base ; for, if it be moved aside in either of the plane faces of the pyramid, it will, § 8, be opposed by the force of restitution due to the action of the molecule of two atoms in the same plane; and if moved out of these planes, its distance from one at least of the atoms in the triangular base must be altered, thus exciting a force of restitution. What has been said of the atom at the vertex of the pyramid is equally applicable to each of those in the base when considered in reference to the three others, and hence the four atoms A, C", G/', D", form a permanent molecule ; and from its capa- bility to resist the approach of a fifth atom, another mole- cule or particle, in every direction, we derive the idea of an elementary solid, having length, breadth and thickness. Elementary solid. A disturbance of any one of the four atoms will put the Dlsturbance wil1 d x cause the neutral others in motion, and it will appear on the slightest con- points to follow sideration that the directions of these motions will be such the dlsturbed atoms. as to cause the neutral positions to shift in the direction of the atoms which have been disturbed from them. 10. What has been said of the action of atoms toSamereasonins applies to molecules and form molecules may easily be shown to be true of the reciprocal action of molecules to form particles, and ofp^"0168- 18 NATURAL PHILOSOPHY. particles to form the bodies which, in all their endless variety of physical characters, come within the reach of our senses. And according to this view, the characteris- tic peculiarities of all bodies are to be understood as aris- ing solely from differences in the action which their atoms, molecules and particles are capable of exerting on each other, and upon those of the bodies with which they may be brought into sensible contact. Eecii.rocai action j>ut jt must |)e remarked that all these differences of confined to small action are confined to small and insensible distances which distances. ^e within the limits of physical contact. At all consider- able distances we find nothing but the action of gravita- tion, of which the intensity is proportional to the number of atoms or the mass directly, and to the square of the distance inversely. The most subtile 8 11. The most subtile and attenuated body of which we body. vl can form any conception, according to Boscovich, is one conceivable; j l o * composed of atoms arranged at distances from each other equal to that which determines the furthest limit of cohe- sion, or that beyond which gravitation begins. But such a body, when abandoned to itself, would shrink into smaller dimensions in consequence of the gravitating force between the atoms not adjacent to each other, and the contraction would continue till the repulsions which it would develope between the contiguous atoms had in- Mk TuT W°Ul(1 crease(l to an equilibrium with the compressing action, permanent form-.when the body would take its permanent form. Such we may suppose to be the constitution of that ethe- Ethcr. reaj me(|ium which pervades all space, permeates every body, and connects us with the objects of wdiose existence we are made conscious through the sense of sight. § 12. A body similarly constituted, but in which the atoms are replaced by molecules or particles arranged at structure of the distances less than the furthest limit of cohesion may 20 NATURAL PHILOSOPHY. Atmosphere, throughout all space, carries this connection to the hea- ether. venly bodies. Disturbance of a The disturbance of an atom, molecule or particle, will rtfLmuled alter its relative distances from the neighboring elements; throughout the molecular forces on the side of the shortened distances space- wm increase, while those on the opposite side will di- minish. The equilibrium which before existed will be destroyed, and the adjacent elements must also be dis- turbed ; these will disturb others in turn, and thus the agitation of a single element will be transmitted through- out space, and impart motion, in a greater or less degree, to the elements of all bodies. Motion affects § 15. Among the bodies thus affected are certain deli- the mind te and net_]ike ramifications of nervous tissue, which through organs of „ m, sense; are spread over portions of our organs ol sense, lnese nerves partake of the agitations transmitted to them from without, and by some mysterious process, call up in the mind impressions due to the external commotion. The structure and arrangement of these nerves differ greatly in the different organs, and while they are all subjected to the general laws which control the corpuscular action of bodies, yet each individual class is distinguished by peculiarities which determine them to appeal to the mind aii our oniy wben addressed in a particular way. We hear, feel impressions due t -i ji i_» t? • • i i.* ~ to a common ancl see ^J tlie operation of a common principle—motion; principle— of this, there is endless variety in perpetual existence among the elements of the media in which we are im- mersed ; and, according as one or another of the organs of sense becomes involved in the particular motion adapted to excite the mind to action, will our sensation become that of sound, light, heat, or electricity. OF WAVES. aii sensations § 16. All sensations derived from our contact with the dependent upon motion. physical world depend, according to this view, upon the ELEMENTS OF ACOUSTICS. 21 state of relative motions among the elements of bodies; and we now proceed to consider those motions which are Those Pr°Per to •i. i i. j xi .• e> it i i produce sound. suited to produce the sensation of sound, and we must be careful to distinguish between the properties of solids and fluids in this respect. Conceive a perfectly homogeneous solid, that is, one in which the particles occupy the vertices of regular and equal tetrahedrons, and suppose its elements in a state of relative repose. A single particle being disturbed from its place of rest, through a very small distance, compared with the tetrahedral edges, will be urged back by the action of the surrounding elements with an energy which is, § 4, proportionate to the disturbance. This particle 0rl)itofa 1 ° r L , r disturbed will, when abandoned to itself under these circumstances, particie; describe about its position of rest as a centre, an ellipse, or perchance, a circle or right line, the extreme varie- ties of the ellipse whose eccentricities are respectively zero and unity. Moreover, the time of describing a Time of complete revolution will, Mechanics, 8 180, be constant, desciiPtion x > 7 o 7 i constant; no matter what the size of the orbit within the limits sup- posed; and the mean velocity of the particle will, there-Mean velocity; fore, be directly proportional to the length of the orbit, or to any linear element of the same, as that of the semi- transverse axis. The disturbed particle being acted upon by its neighbours, these latter will experience from it the action of an equal and contrary force ; they must, there- Neighboring fore, move and describe similar orbits; and the same similarorbi^ will be true of the particles next in order, till the disturb- Disturbance ance becomes transmitted indefinitely. The disturbance ^nsmitted in ^ all directions. must take place in all directions from the primitive source, because the displacement of a single particle from its po- sition of rest breaks up the equilibrium on all sides ; and ,. , -, ,i • • • i • i Disturbance is the disturbance must be progressive, since it is to anpTOgrogsive. actual displacement of a particle that the forces are due which give rise to the displacement in others. It follows, therefore, that while the first disturbed particle is describ- ing its elliptical orbit the disturbance itself is being propa- gated from it in all directions, and that at the instant this 22 NATURAL PHILOSOPHY. First particle having made one circuit, another just begins to move; A third begins to move. Space including particles in aU positions in their orbits; Fig. 10. Illustration. Explanation of wave length. particle has completed one entire revolution, and begins a second, the disturbance will have just reached another particle A2, in the distance, which particle will then be- gin for the first time to move, so that these two particles will during subsequent revolutions about their respective centres always be at the same angular distance from their starting points ; when the first particle A l has completed its second revolution, and the particle A2 its first, the dis- turbance will have reached a third particle A3, still fur- ther in the distance, which begins its first revolution when A2 begins its second, and Ax its third, and so on indefi- nitely. Now, after the disturbance has reached the particle A2 it is plain that the particles between Ax and A2 inclu- sive will be in all possible situations in their respective orbits. For example, taking the instant in which A j first returns to its starting point, it will have described three hundred and sixty degrees, the consecutive particle an arc less than this, the next par- ticle, in order, an arc still less, and so on till wre reach A 2, which will only just have begun to move. If then, we conceive a series of concentric spheres whose radii are re- spectively AlA2, AXA3, AXAA, &c. -*' it is obvious that / J j within the space in- cluded between these spherical surfaces, the particles will be in every possible stage of their circuits around their respective centres, and will, as we pass from surface to surface, be found moving in all possible directions in the planes of their several orbits; and the same would obvi- ously be true, if the radii of any two consecutive surfaces had been increased or diminished by the same length, the only difference being that the particles at the new position Fig. 11. I ELEMENTS OF ACOUSTICS. 23 of the surfaces, instead of being at the origin or places of rest from which they began their respective circuits, would occupy places more or less remote but equally advanced from these points. Thus, for example, had the radii been taken AAA2 + \ A2 A3, and AXA2 + \ A2A2, then Wave length not would the particles at the new surfaces have been at an pea^cculcar ( angular distance from their respective places of primitive position. departure equal to 90°, but the surfaces would still have included between them in the direction of the radii, par- ticles in every possible state of progress in their circuits, the particle at the origin of departure being in this case at a distance from the surface of the smaller of the second set of spheres equal to three-fourths of the difference be- tween the radii of any two consecutive spheres of the first set. This particular arrangement of the particles of any body arising from the disturbance of one of its elements, and by which, after a certain lapse of time, all possible positions around their respective places of rest are occu- pied by the particles, in the order of succession, at the same time, is called a Wave. The distance, in the direction Wave* of the radii, between any two of the consecutive spherical surfaces above described, is called the length of the wave. The term phase is used to express both the par-Phase. ticular displacement and direction of the motion of a par- ticle in any wave. A wave length, therefore, is that interval ^ave length. of space which comprises particles in every possible phase. Particles which have equal displacements and motions, in the same direction, are said to be in similar phases ; Similar Phases- when the displacements and motions are equal and op- posite, the particles are said to be in opposite phases, opposite phases. The surface which contains those particles of a wave which are in similar phases, is called a wave front. Wave frontJ In sound this last term will be used to denote the surface containing those particles which are, for the first time, m sound. be^innin > 7 © #-L remote from the in order, and so on indefinitely, so that the disturbing primitive ^ regulates the value of t, for all particles however remote from the primitive agitation at a. agitation. v independent g i$m With the value of V it is not so; this is indepen- of the disturbing ._ T , n , force. dent of the disturbing force. We have seen, § 12, that when in a state of relative rest, the elements of any me- dium are maintained in that condition by the opposing forces of repulsion between adjacent elements, and of attraction between those which are separated by a dis- tance greater than that which determines the furthest limits of corpuscular action. These forces are equal and opposite. Denote the.sum of the re- pulsions of the particles which occupy Fis-14« a unit of surface by E. Conceive a plane A B, passed through the me- « dium, and the particles on the side JTto be removed ; those distributed x • over a unit of surface of the opposite side will be pressed against the plane by a force equal to E, and to keep the plane from moving would require the application of an equal and contrary force. But this Illustration. B ELEMENTS OF ACOUSTICS. 27 Fig. 15. force, in the case of the atmosphere, is measured by the weight of a column of mercury whose base is unity, den- sity Du, and height h, or by Dit . g . A; whence E= Du .h .(J . . . . . (2). Measure of the elastic force of the atmosphere. The second member measures the Elastic force of the medium. § 20. Let A B, C D, E F, &c, be the positions of several strata of particles of air at rest and of which the molecular forces are in equilibrio ; and suppose them surrounded by a tube whose axis f] | | Demonstration. is perpendicular to their surfaces. If the stratum A B be moved by £JS' any extraneous cause towards the stratum G I), the latter will move under the action of the increased repulsion between it and the stra- tum A B. Suppose the stratum A B to take the position A! B\ at the instant the stratum G D begins to move. The distance A A', will, from the views already given of the constitution of a fluid, be indefinitely small. Denote the distance A G by x; Af C by xt; and the elastic force exerted by the air in its state of rest on a unit of surface by E/ then supposing the cross section of the tube uniform, and its area equal to a, according to Mar- iotte's law a xt : ax:: aE: aE{ Mariotteviaw. in which E/ denotes the elastic force exerted by the air on a unit of surface between A'Br and G D; whence ✓v. Elastic force of TP ___ JP •" J2j — JJj . — . the compressed ^/ air. The stratum C D is urged forward by the elastic force 28 NATURAL PHILOSOPHY. Moving force acting on a stratum; Et, and is opposed by the elastic force E; its motion will therefore be due to x—xt a(E-E) = aE--aE=aE-—-L\ K ' ; X X. Fig. 15. AA' BE' IS J) which is the moving force. And denoting the mass of the stratum C D § by m, the acceleration due to this force, or the velocity generated in a unit of time, will be Velocity due to this force; a E x — x m x. 5 and the velocity v, generated in an elementary portion of time t, equal to that during which the stratum A B moves to thd position A' Bf, will be given by the relation Velocity in small time t\ a E x — x. . v = ■ — • • —— . t, m x. Mechanics §83, which is obviously the velocity with which the stratum G D will be thrust from its state of velocity rest, and is analogous to that imparted to a stratum of fluid imparted to the pressed through an orifice in the bottom of a vessel con- stratum CD. , . n . , \ taming a heavy num. The mass of the stratum GD will be the same whether we regard it concentrated into the plane CD, or ex- panded in both directions half way to the adjacent strata; in which case, its volume would be a. x, and it would have an actual density equal to the mean density of the whole of the fluid mass; for the same being supposed of all the other strata, the matter of the fluid would become con- tinuous. Denoting the mean density of the air under con- sideration by D, we therefore have Mass of a stratum of air. m — D. a . x which substituted in the above equation, and writing ELEMENTS OF ACOUSTICS. 29 therein x for x4 in the denominator, from which it does not sensibly differ, we have E x - x. t Velocitjr v = 1 imparted to J) X X stratum CD; Now at the end of the time t, the stratum A B has reached the position A! B', and the stratum G D begins to move ; that is to say, the disturbance has been propa- gated over the distance from A to G = x, in the time t. Hence, denoting the velocity of this propagation, which is that of the wave motion, by V, we have x 5 t t 1 F = or x ~ V this in the last equation gives E x — x v - Molecular (2)'. velocity; D.V x which may be written -rr x E x — xt D « Here V, is the wave velocity and v, the actual velocity • of a stratum of air, and for the indefinitely small time t, these may be regarded as constant; but the spaces x and x — x( are described with these velocities in the same time, and hence x — xt : x : : v : V whence -rr X — X The same in V = V . ------', X other terms; 30 NATURAL PHILOSOPHY. and this substituted above gives Wave velocity. V3 = -T, E D therefore V = E D (8) whence wTe see, that the wave velocity in the same medium, at a constant temperature and under a constant pressure, will be constant, being equal to tbe square root of the ratio obtained bv dividing the elastic force of the medium by its density. Replacing E by its value as given in Eq. (2), the above reduces to Same in other terms; V = J D . . . (4). Atmospheric density; § 21. The density D, of the atmosphere or any other elastic medium, corresponding to any barometric column /*, and temperature t, is given by Equation (240)' Me- chanics ; that is, by D D h 30m 1 + (t - o2°) . 0,00208 and this substituted in equation (4\ for D, gives Wave velocity for ^iven temperature and pressure. Barometric heisrht V D g . 30in. ±jjl . 11+ (t - 32°) . 0,00208 . . (5). in which Du denotes the density of mercury, and Dt that of the atmosphere at 32° Fah., the atmosphere being under a pressure of 30 inches of mercury. § 22. The quantity h, does not appear in Equation (5); from which we are to infer that the velocity is indepen- ELEMENTS OF ACOUSTICS. 31 dent of the atmospheric pressure, as it should be ; for, an velocity of sound increase of pressure will increase the elastic force E: but ^i?Iw^ ° -•- / atmospiienc this will increase the density D, in the same ratio, so that, pressure; Equation (3), the velocity should remain unchanged. But an increase of temperature under a constant pressure dilates the air, and therefore reduces D for the same value of E Hence, all other things being equal, the velocity greater . , ° ° L in warm weather velocity of sound should be greater in warm than in cold thanincoid. air; greater in summer than in winter, and this is what is indicated by the quantity t, in Equation (5). § 23. If in Equation (5) we make t = 32°, we find F= V 9 • 3°m; ^-.....(6> The density of distilled mercury at 32° Fall, is, Me- chanics, § 275, equal to 13,598, and that of air at the same f. temperature, and under a pressure of 30 inches = 2.5, of Tabularvallies mercury is 0,001301; and the mean value of g is, Mecha-fortbeabove /. data, nics, § 72, Eq. (22), equal to 32,1808, which values in Equa- tion (6) give V= V 32,180S • 255 • i^T- = 915,69 .... (6)' Velocity of sound 0,0013 without increase of temperature. which would be the velocity of sound in our atmosphere under a pressure of 30 inches of mercury and at the tem- perature of freezing water, were it separated from admix- ture with all other media. § 24. But it must be remarked that the value of E, in Equation (3), which is one of the important elements of increase of this estimate, is assumed to be given by the weight due to temperature;or ' & ^ & etherial the height of the mercurial column. 5iow, this only mea- vibration, sures the pressure due to the grosser elements of atmo-pro'lucclby spheric air, and takes no account whatever of the elasticity sonorous waves. NATURAL PHILOSOPHY. Elasticity due to due to that vastly more subtile and refined atmosphere of the ether. ether which permeates the air, glass, and torricellian vacuum, and which, therefore, presses alike on both ends of the barometric* column. A motion among the atmo- spheric strata will give rise to a similar motion in this ether; the equality in its elasticity on opposite sides of the strata in the direction of the motion will be disturbed ; this inequality will develope a reciprocal action among the strata of ether and those of the atmosphere itself; hence, E, in Eq. (3), is too small, and consequently V, is also too small. Denote hjX, a constant co-efficient which, when multi- plied into E, as indicated by the barometer, will give the true elastic force as it actually exists; then will Equation (5) become Corrected for velocity value / ,M' D T ~\ y=\/g.30 .tja-.K. 11 + (*-32°) . 0,00208 I . . (7). or, replacing the value of the first three factors as given by Equation (6)', Velocity as affected by etherial waves, or increase of temperature. Co-efficient of barometric elasticity, K. To find the constant A" T, must be known. V, affected by wind; V= 915,/69 . \/K . (l + (*- 32°) . 0,00208^ . . .(7/. The quantity K, may be called the coefficient of baro- metric elasticity of the air. % 25. To find the value of V, corresponding to any tem- perature t, it will be first necessary to know that of K But K, being constant, if the value of V be found for any particular state of the air, that of If, will result from equation (7)'. The velocity V, is the rate of travel of the front of the wave from a disturbed particle of air taken as an origin. When the wind blows, the whole mass of air, and there- ELEMENTS OF ACOUSTICS. 33 fore this origin, has a motion of translation ; and to find To find v V experimentally, the observations should be so con_experimentaiiy; ducted as to eliminate the disturbing effect of the wind. To understand how this may be done, suppose an observer placed Flg'16, at A, midway between two sta- tions B and G, and the wind to ---------;--------„ blow from B to G. Denote the velocity of the wind by v ; then will the velocity with which sound will travel from B to A, be V + v, and from C to A, it will be V — v, the mean of which is obviously Y. To eliminate therefore the effect of the wind, let four climin°te7.m remote stations B, C, D, E, be so chosen that the line connecting G and B, shall be perpendicular, or nearly so, to that joining E and D, and place an observer at the inter- section A. At the stations B, D, G, E, let signal guns be fired in succession, and the observer at A note, by a stop watch, the intervals of time between his seeing the flash and hearing the report. The distances from A, being carefully measured and each divided by the corres- ponding interval in seconds, will give a value for V. The mean of these values and the reading of the thermome- ter, which must also be noted, being substituted in Eq. (7)', the value of K will result. The experiments of Moll, Vanbeek and Kuytex- Experiments brouwer, performed in 1823, over a distance of 57839 feet, in a dry atmosphere, at the temperature of 32° Fahr., gave a mean value of V— 1089,42 English feet. These values substituted in Equation (7)' give ^(1089,42)* *£*<*" ( 915,69)2 which in Eq. (7)' gives the general value of / Final value of F= 1089,42 Vl+{t-32°). 0,00208.....(8).^^, 3 34 NATURAL PHILOSOPHY. Principle of heat. §26. This vibratory motion among the elements of ether, giving rise to a secondary system of waves, by which the propagation of sound is accelerated, constitutes the principle of heat And to ascertain to what degree a Fahrenheit thermometer would be affected were it sud- denly transferred from a perfectly stagnant atmosphere to one agitated by sound waves, could the mercury take instantaneously the bulk which would enable its ether to vibrate in unison with that of the sound wave, it would only be necessary to find the value of t— 32°, in Equation (5), after substituting for Fand ^/ g m h-~±', their respec- tive values 1089,42 and 915,69. Solving the equation with reference to $'—32°, and introducing these values, we find, Amount of Latent heat rendered free. Difference between computed and observed velocity explained. £-32° = 0,00208 r(i°^y-ii = 199,71 .\ 915,69/ • J This is called the amount of heat given out by an element of air during its condensation in a sound wave. It was to the increased elasticity imparted to air by this sudden change of a portion of its heat from latent to free, that Laplace first attributed the great disparity between the computed and observed velocity of sound. Effect on the stratum CD resumed. 27. Two cases may arise; First case; AA' Before proceeding further we must remark, that nothing has been said of the conduct of the stratum G D, after it was im- pelled forward from its place of rela- tive rest by the action of the stratum A B, which was brought by the disturbing cause, say the motion of a rigid plane, to the position Ar Bf. Two cases may occur : either the stratum A B may be retained in the position A'B', or the disturbing plane may, \>j an opposite movement, leave this stratum unsup- ported from behind. In the first case, if the medium bo BB' ELEMENTS OF ACOUSTICS. 35 homogeneous, the masses of all its particles will be equal, in first case a and the velocity impressed upon those in the stratum ^snfittld in CD will, by the principle of the collision of elastic masses, the direction of be transferred undiminished to those in the stratum EF,the disturbaJlce' after which the stratum C D will come to rest; and the same of the succeeding strata in front: Mechanics, § 247; so that there will simply be a pulse, transmitted along the direction in which the primitive disturbance acted. In the second case, the stratum Af Br, being left unsupported in the second from behind, by reason of rarefaction, will be thrust back- p^J^ * ward by the superior elasticity of the medium in front, transmitted in and this return or backward motion will take place in all direction> the strata in front, in the same order of time and distance from the original disturbance as in the instance of the forward movement; so that a second pulse will be trans- mitted hi the same direction as before, only differing from the first in the backward motion among the parti- cles. Distances § 28. It is easy from the known velocity of sound, to t^vdocityof compute the distance between two places which may be sound; seen, the one from the other; and for this purpose let a gun be fired at one place, and the interval of time between seeing the flash and hearing the report at the other be flamed carefully noted. This interval, expressed in seconds, mul- tiplied by 1089,42 V~T+ (t - 32°). 0,00208, will give the distance expressed in English feet. The value of t will be given by the Fahr. thermometer. Accuracy slightly The accuracy of this determination will of course be a c affected by the wind, should it be blowing at the time. To ascertain the probable amount of this influence, let Abe a sta- Flg*16, tion midway between the places B and G, and suppose the wind---------j--------c to be blowing from B to C, with a velocity denoted by v; denote the distance B A = C A f^^^1 by S, then wTill the actual velocity of sound from B to A, be V+ v, and from C to A, be Y — v; and the intervals 36 NATURAL PHILOSOPHY. Intervals of time of time observed at A, between the flash and report from B and C, will be, respectively, S , and S V+ v V- v or developing these expressions, S First interval; fi — y Second interval; t - S [i_ JL+J^! _&c.l, [ + V + V V ' F2 + « &C •]' Now, the most violent hurricane moves at a rate less than one-tenth that of sound; so that the neglect of the terms involving v2, would in the worst case only involve an error less than Jyth, and in the ordinary cases likely to be selected for experiment their influence would be quite inappreciable, Neglecting these terms, we see that one of these intervals will be just as much too great as the other is too small, and the true interval, denoted by t, will be a mean between them. Hence, Truo interval; Resulting formula for distance. Example. Distance from West Point to Newburjrb. t= - t, + t2 2 S or s = r.t (9). Example. On the occasion of firing a salute of 13 minute guns at ISTewburgh, the mean of the intervals be- tween noting the flash of each gun and hearing the report at West Point, N". Y., was 36,2 seconds; and the temperature of the air, as given by a Fahr. thermometer, was 76° ; required the distance from West Point to New- burgh. S=t. V = 36,2 .1089,42 V1 + (76° - 32°). 0,00208 ELEMENTS OF ACOUSTICS. 37 Computation. S= 36,2 .1089,42 V 1,0915 and by logarithms : 36,2.......1,5587086 1089,42......3,0371954 1,0915, (i), .... 0,0190118 41,202 feet.....4,6149158 5280, feet in 1 mile, ac. . . 6,2773661 7,8034 miles, . . . 0,8922819 » § 29. We have seen that the velocity of sound through VeIocit the air is independent of the barometric pressure, and independent of experiments show it to be sensibly unaffected by its statTofthe0 ° ^ hygrometrical state of moisture and dryness ; the actual atmosphere, ,i i . • i i r» • -i» ,i nature of the weather characterised by log, ram, snow, sunshine ; the ouml &c nature of the sound itself, whether produced by a blow, gunshot, the voice or musical instrument; the original direction of the sound, whether the muzzle of the gun is turned one way or the other; the nature and position of the ground over which the soimd is conveyed, whether smooth or rough, horizontal or sloping, moist or dry. § 30. Eesuming Eq. (7), and denoting by V and P'velocity of sound the velocities of sound through any two gases whatever, by K' and K" their co-efficients of barometric elasticity, and by D' and D" their densities ; then, supposing the barometric column exposed to the pressures of the gases to be 30 inches, and the temperature of the gases to be the same and equal to t degrees, will, Eq. (7), give Vf = \/g.Z&}' ®J±.K' 1+0- 32°) .0,002081 Value in first; D' and V" = \/g^0^^jL . K" I" 1 + (t - 32°). 0,00208J ; Value in second; 38 NATURAL PHILOSOPHY. Dividing the first by the second, we have Velocities compared. H-v/ y rr — V K"' D' (10). Conclusion. That is, the velocities of sound in any two gases, at the same temperature, are to each other as the square roots of their coefficients of barometric elasticities directly, and densities inversely. From Equation (10) we izeadily obtain K' y>* jy K" V"* D" (11). Atmospheric air Taking one of the gases atmospheric air, and the other and hydrogen; hydrogen, and assuming the velocity of sound in hydro- gen, as determined by the experiments of Van Rees, Fea- meyer and Moll, to wit, 2999,4 English feet, we have, after substituting the known values of the quantities in the second member, Ratio of their constant coefficients; Inferenoe; Conforms to Bosco vich 's theory. K" / 2999,4 y .0,0688 = 0,5215, \ 1089,42/ Hence the coefficient of barometric elasticity of air is nearly double that of hydrogen; a result which appears to indicate that the velocity with which sound is propa- gated through gases is in some way dependent upon their chemical or physical constitution. ' This wxmld seem but the natural consequence of the views of Boscovich. VELOCITY OF SOUND IN LIQUIDS. Experiments on § 31. From the experiments of Canton, Oersted, and others, liquids as well as gases are found to be both com- ELEMENTS OF ACOUSTICS. 39 pressible and elastic ; and are therefore fit media for the Experiments on transmission of sound. From the experiments of Colla- pu don and Sturm, on what may be regarded as pure water, Sir John Herschel deduces the compression of this fluid, by one standard atmosphere, to be 0,0000495S9 = e; that is to say, an increase of pressure equal to that arising from a column of mercury having an altitude of 30 inches and temperature of 32° Fahr., will produce a diminution in the bulk of water equal to 10 04090508090 0 0 of the entire volume which it had before this increase. § 32. The quotient e, arising from dividing the diminu-Measure of tion of volume, produced by the pressure of a standard comPressibilit^ atmosphere, by the entire volume before the compression, is the measure of compressibility. § 33. The quotient obtained by dividing the compress- ing force, by the degree of compression, when not accom- panied by a permanent change in the molecular arrange- ment of any body, is the measure of the body's elastic force. Denoting by B, the standard barometric pressure, Measure of T> elastic force. the elastic force E, will be given by E = — . e § 34. Let A B, and CD, be two consecutive strata of water, and suppose the stratum AB, to have been suddenly moved by some Fi»-15- disturbing cause 'to the position A' B'. +*----f----f Denote the distance BD by x, and BD by xt, then, regarding the area of the stratum as unity, will the dif- Illustration; BB' J? ference of volume between A BCD and A'Bf CD, be represented by x — xt, and the degree of compression referred to the original volume, by X — X Degree of • compression: X 40 NATURAL PHILOSOPHY. Compressing force; and the force En necessary to produce this compression will, § 4, be given by the proportion x — xj x ::B:E. in which B = g. Du . h, denotes the pressure due to a stan- dard atmosphere, being the weight of a column of mer- cury whose density is Du and height h. "Whence Its value. ^ B x — x, E = —- ---' 1 e x Combined pressure on a stratum below the surface; But any stratum of water situated below the surface is already subjected to the pressure of the atmosphere, and that arising from the weight of the column of the same fluid above it. Denoting this combined pressure by p, we shall have the stratum A' Bf, and therefore CD, since the resistance to compression arises from the reaction of the latter, urged forward toward EF, by Ed + p ; but the motion of CD is resisted by the pres- Moving force on sure p, whence the moving force becomes E +p — v = E. Ihe distance between A B, -and A' B', being indefinitely small, we may take the mass of the stratum CD, to be measured by D . (x — X,); a stratum; Mass of a stratum; whence the acceleration due to the moving force or the velocity generated in a unit of time, becomes, after substi- tution for B, its value, Velocity generated in a unit of time; E. __ g-i^Dn i D{x—x) ' e.D x velocity in an and the velocity V, imparted to the stratum CD, in an elementary elementary portion of time t, will be given, Mechanics portion of time. ^ /rv , x1 . & 5 x^ncuiius, Ji,q. (9), by the equation ELEMENTS OF ACOUSTICS. 41 Its value; but v= 9-e h.D„ ±. X t X 1* ~ F' which substituted above gives, §33, after clearing the fraction and extracting the square root, Wave velocity in V = \/ 9'h-D" = \/- . . . . (12)water? /. and substituting the numerical values of ^=32,1808; h = 3Qin. — 255 ; Dn — 13,598 ; and denoting by K, what wre Numerical values have before termed the co-efficient of barometric elasti-of data; city, we finally have Resulting wave velocity; v-s/^0^5'13'598-K=me%y/4 (13) K_V 0,0000495S9 . D ' V D in which D must be taken from the table given in §272, Mechanics, corresponding to the temperature of the water. If the temperature of the water be 38°,75 Fahr. D will be unity, and if we assume K — 1, then will /. Velocity when Y = 4696,86. density and constant coefficient arc § 35. A careful and doubtless most exact experimental each equal to determination of the velocity of sound in water was made unity- in 1S26, by M. Colladon. After trying various means for Experiments of the production of sound under water, he adopted the bell, coiiadon; as giving the most instantaneous and intense sound, the blow being struck about a yard below the surface by means of a metallic lever. The experiments were made • ♦ 42 Explanation; NATURAL PHILOSOPHY. at night, the better to avoid the interference from extra- neous sounds, and to enable him to see the flash of gun- powder which was fired simultaneously with the blow. To receive the sound from the water and convey it to the ear, a thin cylinder of tin, about three yards long and eight inches in diameter, was plunged vertically into the water, the lower end being closed and the upper end, to which the ear was applied, open to the air. By means of this arrangement he was enabled to hear the strokes of the bell under water across the entire width of the Lake of Geneva from Kolle to Thonon, a distance of about nine miles. Mean of intervals; § 36. From 44 observations, made on three different days, it appears that the distance of 44249,3 feet was traversed in 9,4 seconds, this being the mean of the intervals between the instant of seeing the flash and receiving the sound at the cylinder, the greatest deviation from which of any single observation not exceeding three-tenths of a second; which gives Velocity of sound in water; 44249,3 ^ ~ 9,4 . . . (14) Four times as ^-Qg making the velocity of sound in water more than great as in air. . four times as great as in air. § 37. The mean temperature of the water, taken at both stations and midway between them, was 46°,6 Fahr. and its specific gravity was found to be exactly that of distilled water at its maximum density, viz.: unity, the expansion arising from the excess of temperature being Experimental just counterbalanced by the superior density due to the determination of saiine contents. This circumstance furnishes at once the the constant K for water; means of finding the numerical value of the coefficient K; for by making D =1, in equation (13), and equating the resulting value of V, and that given above, we have • ELEMENTS OF ACOUSTICS. 43 / 4707,40 y 100m Value of JT; \ 4G96,S6 / ' which differs but little from unity, and from which we infer that there is but little heat developed in the trans- mission of sound through water. And the experiments Inference with ° *■ ivsrard to liquids. hitherto made indicate that this is also true of other liquids. To find the velocity of sound in any liquid it will only be necessary to know its compressibility. A valuable table of the compressibility of different liquids is given by Sir Jonx Herschel, in his Treatise on Sound, Encyc. Met., Yol. 1, p. 770. § 38. In these experiments of M. Colladon, it was found Different tones that the sound of the bell when struck under water if°fs°undin 1 A water and air. heard at a distance had no resemblance to its sound in air. Instead of a continued tone, a short sharp sound was heard like two knife blades struck together; it was only, within the distance of about six hundred yards that the tone of the bell could be distinguished. § 39. M. Colladon also found that sound in water does sound in water not, like sound in the air, spread round the corners ofnotaudiblc , around corners interposed obstacles. In air, a listener situated behind a as in air. projecting w^all or corner of a building, hears distinctly, and often with very little diminution of intensity, sounds excited beyond it. But in water this was far from being the case. When the tin cylinder, or hearing tube, before mentioned, was plunged into water at a place screened from rectilinear communication with the bell, by a wall running out from the shore, and whose top rose above the water, a very remarkable diminution of intensity was heard in comparison with that observed at a point equally distant from but in direct communication with, the bell, or " out of the acoustic shadow" The reason of this apparently singular phenomenon will Acoustlcshadow- appear further on. 44 NATURAL PHILOSOPHY. VELOCITY OF SOUND IN SOLIDS. Solids propagate round better than gases or liquids; Same formula employed; § 40. Solids, when elastic, are even better adapted for the transmission of sounds than gases or liquids. But for this purpose, they should be homogeneous in substance and uniform in structure. The general principle upon which the propagation of sound througli solids depends is the same as in liquids ; and the same formula, Eq. (12), may be employed when the intensity of the specific 7? rrr. elastic force —, § 33, of the solid is knowm. There are, e however, two very important particulars in which they differ. First, the molecules of liquids admit of a perma- nent change of relative position among themselves ; those of a solid are, on the other hand, as before remarked, § 16, subjected to the condition of never permanently altering their relative arrangements without altering their physical character. Second, each particle of a liquid is similarly related to those around it in all directions; while every particle of a solid has distinct sides and different relations to space and surrounding particles. Hence arise a multitude of qualifying circumstances, consequences of ^icb modify the propagation of sonorous waves througli solids, which have no place in liquids, and peculiarities of wave motion become, therefore, possible in the former which are impossible in the latter. Difference in structure of solids and liquids; soius differ from § 41. Solids differ much among themselves in the each other in particulars here referred to. Thus, the cohesion of the molecular L t/y» arrangement; particles of crystallised bodies diners greatly on their different sides, as the facility with which they admit ofj cleavage in some directions and not in others, shows. They have different elastic forces in different directions, Effect upon the an(j fans the velocity of sound through them must de- velocity of . ° sound. pend, .Lq. (12), upon the direction in which the sound is transmitted. A disturbed particle in a perfectly homo- ELEMENTS OF ACOUSTICS. 45 geneous medium becomes the centre of a series of con- centric spherical waves which proceed outwards with equal velocities in all directions. But if the elastic force Mediumnot x t # homogeneous; and density of the medium vary in different directions from the place of disturbance, Equation (12), shows that the shape of the wave front will no longer be spherical. It will be elongated or drawn out in the direction along wave front not which the elastic force is greatest and density least, and sp conversely. Like effects will arise when the elasticity and density both increase or decrease, but unequally. § 42. Thus, conceive a solid whose density, estimated in the direction A C, is represented by D{, and in the direction A B by Dt + c ; and suppose the density to vary gradually from one of these directions to the other, and the law of this variation to be expressed by Fig. IT. Illustration; D = J) (i + o. cos $), (~\K\ Intermediate ' density; in which D denotes the density in any intermediate direction as A D, and the angle which that direction makes with that of greatest density or with A B. Also, let the elastic force in the direction A C be E, and that in the direction A B, be measured by and suppose the elastic force, denoted by E, in the in- termediate direction AD, to be given by the relation Elastic force in direction of greatest density; E = E. 1 — c. cos (16). Same in intermediate direction. Dividing Eq. (16), by Eq. (15), we find 46 NATURAL PHILOSOPHY. Ratio of intermediate density and elastic force; E D E, D 1 — c2 .cos2 ) . (17). Wave front Ellipsoidal. This, since all the factors except 1 — c2 . cos2 (/>, are con- stant, is the polar equation of an ellij)se referred to its centre as an origin; so that the shape of the wave front is ellipsoidal. Velocity of sound in various solids. § 43. By a series of experiments similar in principle to those already referred to, and which it is unnecessary to detail, it is found that the following are the velocities of sound in different solids, that in air being taken as unity, viz.: Tin = 7^ ; Silver = 9 ; Copper = 12 ; Iron (steel?) = 17 ; Glass = 17 ; Baked Clay (porcelain?) = 10 to 12; Woods of various species = 11 to 17. It was found by IIeriiold and Rafk, that when a metal- lic wire 600 feet long, stretched horizontally and held atone end between the teeth, was struck at the other, two dis- tinct sounds were heard; the one transmitted through the wire, teeth and solid materials of the head, to the audi- Dupiication of tory nerves, the other through the air. A similar dupli- cation of sound was observed by IIassexfuats and Gay Lvssac from a blow struck with a hammer against the solid rocks in the quarries of Paris; that propagated through the rock arriving almost instantly, while that transmitted by the air lagged behind. sound. 44. From this it is easy to estimate the time required ELEMENTS OF ACOUSTICS. 47 to transmit the effect of a force applied at one end of a Time re time of with those of the waves which agitate them, the number revolution of a of recurrences of the same condition of these nerves, in a partlclc- given time, will depend upon the length of the waves. The greater or less number of these recurrences deter- mines the character of the sound ; in proportion as this number is greater wTill the sound be less grave or more Acut^ and acute, and in proportion as it is less, will the sound i^e™™8011™13' less acute or more grave. This particular character of 48 NATURAL PHILOSOPHY. Pitch. sound by which it is pronounced to be grave or acute, more grave or more acute, is called the Pitch. Time of cosA+a"cosA") f2 , V^l (a' sin A' +a«*lnJn .cosrgw Jgf 1. displacements; «-f-a — x -M"|_ ^ JT x L*7* \ J and making, supposition; a . cos A = a' cos A' + a" cos A", . . . (a) a . sin A = a' sin A + a" sin A", ...(b) Notation; {jie ai}0ve becomes, after writing d for the total displace- ment, Total displacement; d = -. [cos A . sin (2 tt Y^x\+smA. cos ^ ^f=?) replacing the quantity within the brackets by its equal, viz.: the sine of the sum of the two arcs, we have Same; d = ~ • SHI f 2 tf . Vj~X + a\ . . . (22) x L x J Squaring Equations (a) and (5) and taking the sum, we find, Transformations; a2 = a>2 + a"2 + 2 aV cos {A! - J.") . . . (23) and dividing Eqation (b), by Equation (a), we obtain w , ,. . . a'.smA' + a", sin A" /ft.N deductions; tan A = ------ --- ------ .... (21). # . cos A + a . cos A From Equation (22) we see that the length of the conclusions; resulting wave is the same as that of the partial waves; but the value of A in that equation differing from A\ ELEMENTS OF ACOUSTICS. 65 and An, Equation (23), shows that the maximum dis-Timeof placement for a given molecule does not take place ^p^ent in with the same value of t, as for either of the compo-resulta^twave' nent waves. The maximum displacement —, which determines the x intensity of the sound, in the resultant wave, is given by Equation (23) to be - = I . /a'2 + a"2 + 2a'a".CM{A'-A") . . (25) General value of ® X \/ ... which depends upon the arc A' — A". Its greatest va- lue is obtained by making Af — A" = 0, in which case we have Fig. 23. this displacement; When this value is greatest; a af + a x x tr Greatest value; its least value results from making A' — A" = 180°, in which case a x a' - a" x Fig. 24. In the first case Equation (24) gives tan J. = <*' + a") ■ sin^l _ tan A' {a'+ a"), cos A' = tan J"; When this value is least; Least value; First case; whence A, is equal to Ar, and to A", and the maximum conclusion; displacement will occur at the same place and at the same time in the resultant wave, and in both compo- nent waves. In the second case, if we substitute in Equation (24) Ar = 180° + A", we find 66 NATURAL PHILOSOPHY. Second case; Conclusion; tan J. = (^O^1^ =tan^L"=tm(180o+ Af)=tnnA'; (a"-af) cos A" that is, A is equal to one at least of the arcs A' and A", and the greatest displacement in the resultant wave will occur at the same place and time as in one of the component waves. intensity of § 59. If the intensity of soundsupposed somic[ jn the component component waves be supposed equal at waves; the place of superposition, a then will a! = a", and Eq. (25) becomes Fig. 25. Consequence; a x 2a/_ x cos A' - A" 2 (26) and Equation (24) reduces to Reduction; , A sin A! + sin A" . A' + A" tan A =----------— = tan------ cos A' + cos A" 2 Value of arbitrary constant; or. A = A' + A" 2 When A' - A" = 0, then will Eq. (26) give (27). Supposition; °L = ^°L, and A = A' = A"; x x Illustration. that is, the intensity of sound in the resultant wave will be double that in either of the equal component waves; and the greatest displacement will occur at the same time and place in the component and resultant waves. Fig. 26. ELEMENTS OF ACOUSTICS. 67 If a! and a" continue equal, and we make A!—J."=180°, supposition; then will Equation (26), give a - = o; X (26)'. Consequence; or in words, one of the equal silence sounds will destroy the other. Fi& 2T' produced; Thus it appears that two equal sounds reaching the same point may be in such relative condition that one will wholly neutralize the other, and the two produce perfect silence. This phenomenon is called the Inter- Interferen<* of ,. r» t sound. jerence ol sound. With any other values for A' and A" than those which give Ar - A" = 180°, Equation (26), shows that a . 2a' - <—; X X Result of partial coincidence of two sound waves. that is, that the sound in the resultant wave will be less than double that in either of the equal component waves. 60. To ascertain the precise relation between twoConditions tlia^ i . i .-n t -,, will cause two equal waves, which will cause one to destroy the other, equai waves t0 neutralize each other. make, 'in Equation (20), A' = A!' ± 180° = A!' ± « and we have a' . r v.t—x aii i df = -r sin [2* —^— + A" db «\ but 7T — 2*. X 2. X Transformations; 68 NATURAL PHILOSOPHY. and this substituted above, the equation becomes Resultant displacement; * a' • To^ V.t-x±\-k A„i d— — . sm 2tf.-------------\-JL x' L X J Fis. 27. Conditions in words. which becomes identical with Equation (21) by writing x, for aj±}X, That is to say, one wave will destroy another of equal length and intensity, provided the first be in advance or in rear of the second, by half the length of the wave. And since the retardation of a whole length of a wave, two whole lengths, three whole lengths, &c, produces no alteration in an undulation, it is obvious that a retardation of _-, -^->-^->&c-5 w^ Jj 2i 2i produce the same effect as a retardation of —: and thus two waves will destroy each other if the maxima of vi- bration be the same, the waves of the same length, and transmitted in the same direction ; provided, one follow the other by one half, three halves, five halves, &c, when waves of a wave length. If the waves do not proceed in the interfere only at same direction, they will interfere only at the point of pomt o umon. ^^^ ^xq above conditions being fulfilled. Same considerations applicable to three or more equal waves; § 61. The same process of combination may be ap- plied to three, four, &c, waves of equal lengths. Thus let there be the Equations d' = - sin [2* x L ■•] Equations to be used; *"=£■'Bin [2* Z*Z? + A"] ELEMENTS OF ACOUSTICS. 69 rf"' = ^sin \2«.I±?+A"'] X L A. J Operations performed; Pis. 28. adding these, developing the sine of the sum of the two arcs within the brackets, col- lecting the common factors and denoting the resultant displacement by d, we have ^=-cos^.sin2*.Z^ + lsin^.cos2tf.Z^ ■n Illustration; X x -X X X Resultant displacement; or a = — sin X 2«. - V .1 — X in which + ^J; The same; a cos A = a' cos A' + a" cos A" + a'" cos A'" = X a sin A = a'sin A' + a" sin A" +. a'" sin J/" = T a — T ^/X2+T2; to0LA=-g- Notation. Fig. 29. § 62. Although it is possible for two waves of sound, whose lengths are the same, to neutralize each other, it is not so when the waves have un- equal lengths; for, Eq. (22) u was deduced by making V and X the same in the two component waves, the sum of dr and dtf being in that case reducible. If these con- ditions were not fulfilled, this sum would not be reducible, and there would be the two arcs Two unequal waves cannot neutralize each other; Illustration; 70 NATURAL PHILOSOPHY. Explanation; *k----:---, and 2 *-------, \' X in the final value for $, with different coefficients, which could not be made equal to zero at the same time. The values of V, will, to be sure, be the same in any two waves of sound, but this need not be so with those of X ; and in waves which produce light, in which subject we shall have most occasion to refer to the doctrine of in- terference, the values of F, as well as those of X, may differ. The discussions of waves of different lengths may, unequal waves; therefore, be kept perfectly separate, as the combined effect of such waves will be the same as the sum of their separate effects, without the possibility of their destroying or modifying one another. Conclusion respecting NEW DIVERGENCE AND INFLEXION OF SOUND. Any disturbed particle causes subsequent disturbance in another; Same true for all particles in a wave front: Illustration; Fig. 30. § 63. We have seen that every disturbance of a mole- cule at one time is truly a cause of disturbance of an- other molecule at some subsequent time. All the mole- cules in a wave front become, therefore, simultaneously centres of disturbance, from each one of which a wave proceeds in a spherical front, as from an original dis- turbance of a single molecule. Thus, in the wave front A B, a molecule at x becomes a new centre of dis- turbance as soon as the wave front reaches it; and if with a radius equal to V.t a circle be described, this circle will represent a section C of the spherical wave front proceed- ing from x, with the velocity V, at the end of the interval of time de- noted by t And the same being true for the molecules x', x", &c, of the primitive wave, there will result a series of intersecting circles ELEMENTS OF ACOUSTICS. 71 a particle; having equal radii, and the larger circle A' B1 Construction of . ° resultant wave tangent to all these smaller circles, will obviously be a front; section of the main wave front at the expiration of the interval t, after it was at A B. Any molecule situated at the intersection of the smaller circles will obviously be agitated by the waves transmitted to it from mole-Eesultant cules at their respective centres ; and the resultant dis- displacement of placement will, §55, §56, be the algebraic sum of the displacements due to each when superposed. Hence, to find the disturbing effect of any wave upon a given molecule at a given time, divide the wave into a number of small parts, consider each part as a centre of disturbance, and find by summation the aggregate of all the disturbances of the given molecule by the waves coming from all the points of the great wave. The cause which makes the disturbance of a single molecule at one instant the occasion of the simultaneous disturbance of an indefinite number of surrounding mole- , 1 . ° Principle of new cules at a subsequent instant, is called the principle of divergence new divergence, of which frequent use wTill be made instated; the subject of light. Eule. aiV § 64. Let us trace the consequences of this principle Its application to in its application to the passage of sound through aper-soundthTugh tures and around the edges apertures and Of Objects. Take a parti- ri* «• around corners; tion M JS~, through which there is an opening A B, and suppose a spherical wave of sound to proceed from a centre 0. Only that portion of the wave which comes against the opening can pass through, and the wave front on the opposite side of the partition will be found by taking the diffe-Illustration» rent points of the segment A B, within the opening as centres, and radii equal to V. t, and describing a series of elementary arcs, and drawing a curve tangent to them ?2 NATURAL PHILOSOPHY. Explanation and construction; Fig. 81. all. That portion of this tangent curve included be- tween the lines C A' and C' B', drawn from C, and tangent to the limits of the opening, will obviously be p.. 2\£MlFtIM. Sound that is not reinforced by particles from the primitive wave; Sectors wherein the sound is due to superposition of waves from the edges; Intensity- increased by coincidence; Decreased by interference. Points taken between the edges; the arc of a circle having C for its centre. The elemen- tary circles described about the limits A and B as cen- tres, cannot be intersected at points exterior to the angle Ar CB! by those described with equal radii from points of the wave front lying between A and B; the wave front within the angles A! A M and B' B N~, will have their centres at A and B respectively; and the sound proceeding from these points will be diffused over the arcs A' M and B' N without reinforcement from molecules of the same primi- tive wave. But other waves from C reaching the opening in suc- cession, a spherical wave diverging from B, and of which the radius is B 0, will be overtaken by a subsequent one from A, having for its radius A 0 ; so that, the intensity of sound in the angle A' A M will result from the super- position of the disturbances from B and A. The same will be true of the sector Bf B JV. Now, if B 0 - A 0, be equal to X, 2X, 3X,.. .^x, in which n is a whole number, then will the intensity of the sound be increased above that due to either of the com- ponent waves. But if B 0- A O, be equal to \\, f X ---(n + i)\ n being still a whole number, the compo- nent waves will interfere at 0, and the intensitv of the sound will be lessened at that point by the prevention there of the disturbance due to either of these two component waves. Taking another molecule Bt, nearer to A, the wave from Bn will interfere with the wave from A, but at a point <9y, nearer to the partition ; in order to pre- serve the difference BtOt-A Ot, the same as before, ELEMENTS OF ACOUSTICS. 73 to wit, (n + i) X. Assuming other points in succession construction; nearer to A, we shall find the interference to take place at molecules still nearer to the partition; and finally, when we come to a molecule Q, in the main wave front whose distance A Q, from A is equal to (n +1) X5 the interference will occur at a molecule situ- ated against the partition at P. Now, making n = o, in which case A Q will equal \ X, and applying \ X from a to aJ, tbp wnvPS from n, p~ a l c a,*, c *______q:................. Illustration; tilt? WdVUD JLLUIII CO lU.lilniUUI.iSiiiliiMiilll •---■----—------■---•----■ » »------ IIIIIIIII.IIMi^Hlllllllillllllllil and a4, will inter- fere at P. Applying \ X, from b to bd, the waves from b and b4 will also interfere at the partition; and in the same way it may be shown that all the partial waves from molecules in the distance A Q, will interfere with those from the molecules in the distance Q D, Q D being equal to A Q. Commencing the same process at D, we see that the opening may be such that on applying \ X from a' to a\ , this latter point art may be in the posi- Explanation of tion from which there can be no new divergence to inter-results; fere with that from a!; and the same for the whole of the arc D B, of the main wave. This latter is, therefore, left, as it were, undisturbed, and sound from it may or may not be audible at P, depending upon the extent of this arc and the intensity with which the sound reaches the opening. The distance A Q is equal to \ X. But X, we sound heard at have seen, 8 48, varies with the pitch, whence the sound Partition dePenfl8 ' u 7 x ' on pitch, and size heard at P, will depend upon its pitch and the size of opening. of opening through which it may pass. § 65. From what precedes we see that at the line Acoustic shadow A 0, Fig. 31, there begins, as it were, an acoustic shadow,c which deepens more and more as we approach the partition towards P, where the sound becomes leastInflexion of audible. This bending of sound around the edges of an opening is called the Inflexion of sound. 74 NATURAL PHILOSOPHY. Case of sound bending around corners; Explanation; No perfect neutralization; Grave sounds more audible than the acute; Case of little inflexion. ffH.iiMiiiimiiiiii»iiiHiiiiiil^ Fig. 83. Qn JO M JD'H, D.* § 66.When the opening is continued indefinitely in one di- rection only, we have the case of sound bending around a cor- ner. But when the open- ing is continued indefinite- ly in one direction, there can be no arc of the main wave as D B, (last figure), without a corresponding arc Dd Bt, further on, to neutralize it in part at least by interference, and hence, were the component sounds of the same intensity at the point of superposition, they would produce perfect silence, and no sound could be heard at P. The sound from the main wave is of the same intensity throughout on reaching the corner ; the new diverging waves leave their respective centres, which are distri- buted along the front of the main wave, with equal intensities ; they can only interfere after having travel- led over routes which differ by |X; the intensity of sound varies inversely as the square of its travelled dis- tance ; and the intensities cannot be equal at the places of interference, and therefore can only partially neutralize each others' effects. This is shown by Equation (26)', in a which _ is zero, only because x, under the conditions x there imposed, is the same denominator for a' and a". In sound, X varies, as we have seen, § 48, from a few inches to many feet, and as the difference of intensities in the interfering waves will be greater as X is greater, the graver sounds would be heard, under the circumstances we have been considering, more audibly than the more acute. If the lengths X, were insensible in comparison with the route travelled, there would be but little in- flexion ; since, in that case, the intensities of sound in the interfering waves would be sensibly the same, and it would require but a slight obliquity from the direct course of the main wave to make a difference of route B 0 — A 0, Fig. 31, equal to \\ necessary for one wave sensibly to destroy the other. ELEMENTS OF ACOUSTICS. 75 Fie. 34 Person behind a wall listening to a band of music; An auditor placed behind a wall at P, would hear the bass notes from a band of music playing at a position A on the opposite side, much more distinctly than the acute notes. At P, the notes of the tuba, for instance, might be heard distinctly, while those of the octave flute would be lost to him. In passing from the position P to 0, he would catch in succession the higher notes in order of the ascending scale, and finally, when he attained a position near the Position wnence direct line A 0, drawn from A tangent to the corner, he a11 the would hear all the instruments with equal distinctness, if \qv^7u&ib\l played with equal intensity and emphasis. The facts and explanations here given have an important applica- tion in the subject of optics. If we suppose the lengths of sonorous waves propagated through water to be much shorter than those through the air, we have here a full and satisfactory explanation of the phenomenon observed by M. Colladon, mentioned at the close of § 39. Indeed, taking the acoustic shadow Foregoing there referred to as established, it must follow as a conse- dcd*ctlons ' conformable to quence, from the principle of new divergence, that the experiments. lengths of the sonorous waves in liquids are shorter than in air. REFLEXION AND REFRACTION OF SOUND—ECHOS. 8 67. There is no body in nature absolutely hard and *> J J Disturbed inelastic. "Whenever, therefore, the molecules of a vi-particles of one brating medium come within the neutral limits of those body agitate # those of another, forming the surface of any solid or fluid, they will and transmit a agitate the latter with motions similar to their own, and pulse- a pulse will be transmitted into the solid or fluid with a velocitv determined by its density and elastic force. 76 References; NATURAL PHILOSOPHY. § 68. Referring to the transmission of sound through air, and resuming Equation (2)', we have, after substituting the value of V, as given by Equation (3), Velocity of a particle; x — xt X Now, by reference to § 34, it will be seen that Excess of condensation; X — XJ X expresses the excess of condensation on one side of a molecule over that on the opposite side. Making Expressed by an equation; X >-=c, the above Equation may be written Velocity of a particle; v= G D (28). In the same homogeneous medium E and D are con- fute for stant, whence we conclude that the actual velocity of a K.mogeneous mojecuie which is the same as that of the stratum to media. " which it belongs, is directly proportional to the excess of condensation on one side of it, over that on the oppo- site side. wi.en a particle "When, therefore, by the forward movement of a mole- will come to rest . . . -, cule the condensation becomes equal on opposite sides, the molecule comes to rest, and remains so till again disturbed by some extraneous force. This explains why it is that a pulse transmitted through a medium of uni- ELEMENTS OF ACOUSTICS. 77 form density sends back no disturbance, but leaves every Livinsforce molecule behind in a state of rest. The living force im- rans pressed upon any given stratum is transferred to the next one in front, and this to the next in order, and so on in- definitely. Fig. 35. jb e' a w X, J}' KK' W A A! J) zr x. A pulse transmitted in § 69. When the stratum , j r> • Fig. 35. disturbed; stratum A B is mo- ved by some source of disturbance to A! Bf, the stratum CD will move in the same direction, and a pulse will be transmitted on-direction of , disturbance; ward towards W, the excess of condensation being on the same side of the moving stratum as the place of the ori- ginal disturbance. But a shifting of the stratum A B to the position A! B', leaves the excess of condensation And also one in which acts on the stratum C I)f on the opposite side °ppo^to 1 L direction; from A B / the - stratum C D' will therefore close in upon A' Br, and the same occurring in succession with all the strata on the side towards W, a pulse will be trans- mitted in an opposite direction from that which begins with the motion of CD. Thus, every case of an original disturbance of a molecule will give rise to two pulses pro- Eveir -i. . .. -i. .. 'i-iii i •, ,i disturbance pro- ceedmg m opposite directions, with the same velocity, the ducestwopulses; two pulses differing only in this, viz.: in the one the wave velocity will be in the same direction as that^)fTheirdifference- the molecules, and in the other in an opposite direc- tion. § 70. The elastic force E, of two media in contact and Elastic force of at rest, must be the same; otherwise motion would ensue. Con^tandat When, therefore, in the progress of a pulse, it reaches rest- a stratum X Y, of a density different from that of those which precede it, Equation (28), shows that for the same excess C, of condensation, the velocity of the stratum will Effect when the be altered; that is, the actual motion of the molecules ^11" will either be accelerated or retarded. If the new stra-greater density; • 78 NATURAL PHILOSOPHY. turn be of increased density, the next preceding stratum KI, will be checked in its progress by the greater mass of X Y, and brought to rest before it reaches its neutral distance from that behind; the excess of elastic force thus retained will react upon the next preceding stratum which has already come to rest, and will thus give rise to a return pulse in which the velocity of propagation and that of the molecules will be in the same direction. Effect when jf on fae contrary, the new stratum have a dimin- moving stratum , meets one of less ished density, the motion of K L will be accelerated, the density. density in front of the next preceding stratum will be- come less than that between those behind which have come to rest; these latter strata will therefore move for- ward in succession, and thus a return pulse will be pro- duced as before, but with the difference, that the velocity of propagation and that of the molecules wTill be in oppo- site directions. wave meeting a § 71. It follows, therefore, that when a pulse or wave medium of 0f g0und in any medium reaches another medium of different density , , . . . is resolved into greater or less density, it is at once resolved into two, two; one of which proceeds on through the second, while the other is driven back through the first medium. cause of this This division of an original pulse into two others, arises resolution. entirely from the reciprocal action of the two media on each other. If the media be perfectly elastic, there can be no loss of living force, and the sum of the intensities of sound in the component pulses will be equal to that of the original pulse. If the media be not perfectly elastic, there will be a loss of living force, and the sum of the intensities of the component pulses wrill be less than that of \kxe original pulse. incident, The original pulse is called the incident; that transmit- reflecTed^L. ted into tlie sec°nd medium, the refracted; and that driven back through the original medium, the reflected pulse. To an ear properly situated, the reflected pulse will be audible, and is, for this reason, called an echo. The sur- Echo. ELEMENTS OF ACOUSTICS. 79 face at which the original pulse is resolved into its two Aviating component pulses, is called the deviating surface. § 72. To find the law which regulates the direction of Direction of the the reflected pulse; let A M be a ^ ^ ^^£* portion of the front of an incident spherical pulse, so small that it may be regarded as a plane. Draw 31 A", Ar i\T and A 0, normal to the pulse, and suppose the latter, moving in the direction from iVto A', to meet the face E G of a second me- dium. Each molecule of the pulse as it recoils from the surface E G, becomes the centre of a diverging spherical pulse which will, Eq. (28), be propagated with the velocity of the incident pulse. Accordingly, when the portion 31 Explanation and reaches the face of the second medium at A", the por-c tion A will have diverged into a spherical pulse whose radius is A B = A" M. In like manner, if A' 31' be drawn parallel to A 31, the portion diverging from A! will, in the same time, have reached the spherical pulse whose centre is A! and radius A! Bf = Af N. The same construction being made for all the points of the incident pulse as they come in succession to the deviating surface, the surface which touches at the same time all these spherical surfaces will obviously be the front of the re- flected pulse. But because A' B' and A B are respec- tively proportional to Af JV and A" M, and as this is true for any other similar lines drawn from points of the deviating surface to the corresponding points of the in- cident and reflected pulses, this tangent surface is a plane. Inci(ient and Moreover, since A B is equal to MA", and the angles reflected pu1sos A MA" and A" B A are right, the angles MA A" and aTgieVvvith BA" A are equal, and the incident and reflected pulses deviating surface. make equal angles with the deviating surface. Any line which is normal to the front surface of a 80 NATURAL PHILOSOPHY. Ray of sound. Angle of incidence; Angle of reflexion; These angles equal. Fig. 87. pulse, is called a ray of sound. The angle NA D, which the normal to the incident pulse makes with the normal to the deviating surface, is called the angle of incidence. The angle BAD, which the normal to the reflected pulse makes with that to the deviating surface, is called the angle of re- flexion; and because the angle made by two planes is equal to that made by their normals, we conclude from the foregoing, that in the reflexion of sound, the angles of incidence and of reflexion are equal. Direction of the § 73. The law which determines the course of the re- actennineT-8* fracted pulse is equally simple, and is deduced in a man- ner analogous to the preceding. Let A 31 be an inclined element- ary plane pulse, incident upon a de- viating surface E G, at any instant. In the interval of time during which the point 31 is moving from M to A", the agitation which begins at A will have reached some spherical surface within the second medium of which A B is the radius ; and in like man- ner, the agitation which begins at A', will have reached some spherical surface of which A! B' is the radius, by the time the portion of the inci- dent pulse at M, will have passed on to A"; and the same of intermediate points of primitive disturbance on the deviating surface between A and A", the first and construction and last points of incidence. The surface tangent to all these spherical surfaces will be the front of the transmitted or refracted pulse ; and because A B and A' Br are respec- tively proportional to A" Mand A!N, this surface is a plane. ELEMENTS OF ACOUSTICS. 81 • The angle JST AD, made by the normal to the refracted pulse and that to the deviating surface, is called the angle of $ refraction. Denote the angle of incidence NAD, which is J)~ equal to the angle 31 A A", Fig. 38, by 9 ; and the angle of refraction N' A D, which is equal to the angle A A" B, Fig. 38, by 9'; then will MA" = A"A. sir AB = A"A. sin and dividing the first by the second MA" sin 9 A B sin 9' ' but A" M and A B, being described in the same time, Explanation; the first by the incident, the second by the transmitted pulse, are respectively proportional to the velocities in the two media. Denoting the velocity of the incident pulse by V, and that of the transmitted pulse by V\ we have F_ _ MA" _ sin 9 V ~~ A B ~ sin 9' whence sin 9 — —T sin 9', .... (29). That is to say, in the refraction of sound the sine of the Rule. angle of incidence is equal to the sine of the angle of re- fraction multiplied into the ratio obtained by dividing the velocity before incidence by that after refraction. 6 Fis. 39. Refracted sound; Illustration; up; Ratio of velocities of incident and refracted sound; 82 NATURAL PHILOSOPHY. Application to air and water; Illustration; Thus, if sound proceed through the atmosphere at 32° Fahr., and be incident upon the surface A B, of water at /- the same temperature, then will V = 1089,42, V'= 4707,4, and V= 1089,42 V 4707,4 which in Eq. (29) gives sin 9 = 0,23142 . sin i iii- Afferent different distances from a speaker may throw back to him distances reflect numerous echos of the same sound. Of this many re- manyechos of __ same sound; markable instances are recorded. At Lurley-Fels, on the Rhine, is a position in which a sound is repeated by echo seventeen times. At the Villa Simonetta, near Milan, is another where it is repeated thirty times. An echo in a building at Pavia used to answer a question by repeat- ing its last syllable thirty times. The rolling of thunder has been attributed to echos from clouds situated at unequal distances from an auditor; and the propriety of this view has been sustained by the observations ofSeveral # remarkable Arago, Matthietj and Proney, while experimenting upon instances; the velocity of sound. They found that when the weather was perfectly clear the reports of their guns wTere always single and sharp; whereas when the sky wras overcast or a single cloud of any extent was present, they were fre- Experiments; quently accompanied with a long continued roll like that of thunder, and occasionally a double sound would arrive from a single shot. But it is proper to remark that the rolling of thunder admits of another explanation. Thunder is caused by a disturbance of electrical equilibrium in the atmosphere; experience shows that this takes place over a long and sinuous line, the different points of which are at unequal Eoiiing of distances from the auditor, and the sounds from these thunder- points can, therefore, only reach him in succession and without sensible intervals. § 82. When reflected sound and that proceeding di- rectly from the same source, are made to fall upon the 90 NATURAL PHILOSOPHY. Reflected sound may increase the effect of direct sound; Fig. 48. Illustrated by the speaking trumpet; R^'N M ^M Its construciion and use explained; By its use sounds are rendered audible that could not be heard without it. ear simultaneously, or nearly so, they strengthen each other and become audible in positions where neither could be heard separately. The Speaking Trumpet affords an illustration of this. The Speaking Trumpet is a funnel-shaped tube, of which the object is to throw the voice beyond its ordinary range. In its best form it is parabolic. It is a geometrical property of the parabola that a line FT, drawn from the focus F, to any point T of the curve, and another TK, drawn from T parallel to the axis FA, make equal angles with the tangent line to the ^*\ O "v. curve at T. A portion of the diverging rays of sounds proceeding from a mouth at the focus F, will be reflected by the trumpet in directions parallel to the axis A F; and the living forces of the aerial molecules which, without the trumpet, would have been diffused over that portion of the spheri- cal surface on the outside of a cone of which F R and FR are the most diverging elements, become, by its use, concentrated within the limits of a circle whose diameter MX, is equal to that of the trumpet's mouth, and superposed upon the living forces arising from the action of the direct sound. The axis of the trumpet be- ing directed upon a person at a distance, sounds of audi- ble intensity may thus be conveyed to him, which he could not hear from the unassisted organs of speech. § 83. The Hearing Trumpet, which is intended to assist per- sons who are hard of hearing, is similar to the speaking trum- HearingtrumPet;Pet; but tlie operation is re- versed. The rays of sound en- ter this instrument at the larger Fig. 49. ELEMENTS OF ACOUSTICS. 91 opening and are so reflected as to become united at the construction smaller end, which is inserted into the ear. and use; § 84. Whispering Galleries, so called from the fact whispering that the faintest whisper uttered at one point may be dis-gallerie8; tinctly heard at another and distant point, without its be- ing audible at intermediate positions, depend upon the operation of the same principle, to wit, the convergence of the rays of sound by reflexion. The best form forBest form- these galleries is that of the ellipsoid of revolution. In such a chamber two persons, one in either focus, could keep up a conversation with each other which would be inaudible at other points. The ear of Dionysius is Ear of Dionysius. celebrated in ancient history; it was a grotto cut out of the solid rock at Syracuse, in which a person placed at one point could hear every word, however faintly uttered, in the grotto. It was doubtless of a parabolic shape. The same principle is employed in the construction of Speaking Tubes, used for the purpose of communicating between different apartments of the same building, now coming into very general use. Fte. 50. Speaking tubes on same principle. Halls for public speaking; § 85. Halls for public speaking, such as lecture rooms, theatres, churches, and the like, should be so constructed as to diffuse the sounds that are uttered throughout the space occupied by the audience, unimpaired by any echo or resound. Were the speaker to occupy constantly the same position, the parabolic form would,on theoretical grounds, undoubtedly be the best; but in debating halls, where every speaker occupies a dif- ferent j>osition from another, these conditions are very Principles on difficult to fulfil, especially when the room is large. Every- T1"^ !1,ey . ° J should be thing should be avoided that would at all interfere with constructed. 92 NATURAL PHILOSOPHY. Experiment; Illustration; Explanation. the uniform diffusion of sound, and especially all need- less hollows and projections which are likely to gene- rate echos. The following experiment will illustrate, in a very simple manner, the consequences arising from the re- flexion of the rays of sound from the interior of a pa- rabola. Place a watch in the focus A of a parabolic mirror 31N, and all the rays of sound that fall on the concave surface will be reflected in the direc- tion indicated by the arrows. The ticking of the watch will be plainly heard within the space MN 0 P, in which the rays fall, but it will not be audible at a small distance on either side. Now place a second reflector 0 P, opposite to the for- mer, and at some distance from it; the rays of sound will be received by it and thrown into the focus B. If the ear, or, better still, the mouth of a hearing-trumpet, be applied to this point, the ticking of the watch will be heard as plainly as at A. § 86. While it is important to diffuse sound uttered, Pai-tition walls; or in any way produced uniformly, so as to render it distinctly and equally audible in all directions, it is also necessary to prevent its passage from one apart- ment to another for which it was not intended. Parti- tions are usually made of solids ; but solids, if elastic, such as wood, metals, and stone, are, as we have seen, better adapted to transmit sound than air itself; an essential condition, however, for this transmission is homoge- neousness of substance and uniformity of structure. Where How constructed these are wanting a sonorous pulse transmitted through Amission of a so^ *s ever changing its medium, and soon becomes sound. broken up by reflexion and refraction, retardation and ELEMENTS OF ACOUSTICS. 93 acceleration, into a multitude of non-coincident waves, Examples of interference 01 and these from the laws of interference must, to a greater B0Und. or less extent, destroy each other. As an instructive instance of this stifling effect on a sonorous pulse, we may mention the example afforded by a tall glass filled with champagne. As long as the effervescence lasts and the vdne is full of bubbles, the glass cannot be made to ring by a stroke on its edge, but will give a dead and puffy sound. As the effer- Glass of vescence subsides the tone becomes clearer, and whenc the liquid is perfectly tranquil, the glass rings as usual. On re-exciting the bubbles by agitation, the musical tone again disappears So of a solid or union of several solids, in which Heterogeneous there are frequent changes of density and elasticity,soIlds; and especially where there is a want of adhesion among the different parts; sound penetrates these with great difficulty, and materials so united as to satisfy to the greatest extent possible the condition of non-homogeneous- ness should, therefore, be employed whenever it is an object to prevent the transmission of sound. The influence of carpets, curtains, and tapestry hangings, in preventing reflexion and echos in large apartments, is due to the causes above mentioned. The mixture of the unelastic carpets, curtains, fibres of the cloth with its numerous layers of entangled &c- air, intercepts and deadens the sonorous waves before they reach the more solid and elastic media behind. 9i NATURAL PHILOSOPHY. MUSICAL SOUNDS. Audible sounds produced; Impression on the ear depends upon; Auditory nerves analyze pulsations: whence grave, acute, &c. sounds; and tones of musical instruments. Noise; Crack; Crash; § 87. Every impulse mechanically communicated to the air or other elastic medium is, as we have seen, propagated onward in a wave or pulse; but in order that it may affect the ear as an audible sound, a cer- tain force and suddenness are necessary. The slow wav- ing of the hand through the air is noiseless, but the sudden displacement and collapse of a portion of that medium by the lash of a whip, produces the effect of an explosion. The impression conveyed to the ear will depend upon the nature and law of the original impulse, which being altogether arbitrary in duration, violence and character, will account for all the variety observed in the continuance, loudness and quality of sound. The auditory nerves, by a most refined delicacy of mechan- ism, appear capable of analyzing every pulsation, and of appreciating the laws which regulate the motions of the molecules of air in contact with the ear; and from this arise all the qualities—grave, acute, harsh, soft, mellow, and nameless other peculiarities which we distinguish between the voices of different individuals and different animals, and the tones of different musical instruments —bells, flutes, cords, &c. § 88. Every irregular impulse communicated to the air produces what we call noise, in contradistinction to musical sound. If the impulse be short and single, we hear a crack; and as a proof of the extreme sensibility of the ear, it is to be remarked that the most short and sudden noise has its peculiar character. The crack of a whip, the blow of a hammer against a stone, the explo- sion of a pistol, are perfectly distinguishable from each other. If the impulse be of sensible duration and irre- gular, we hear a crash; if long and interrupted, a rattle, ELEMENTS OF ACOUSTICS. 95 or a rumble, according as its parts are less or more con- Rumble. tinuous. § 89. The ear retains for a portion of time after the Continuoug impulse is communicated to it a perception of excitement, sound produced; If, therefore, a short and sudden impulse be repeated beyond a certain degree of quickness, the ear loses the intervals of silence and the sound appears continuous. The probable frequency of repetition necessary for the production of continuous sound is stated to be not less than sixteen times in a second, though the limit will be The frequency of ' ° repetition different for different ears, necessary. § 90. If a succession of impulses occur at exactly equal Musical sounds; intervals of time, anft if all the impulses be exactly simi- lar in duration, intensity, and law, the sound produced is perfectly uniform and sustained, and takes that pecu- liar and pleasing • character called musical. In musical sounds there are three principal points of distinction, viz.: the pitch, the intensity, and the quality. Of these JaJfYahT^7 the pitch depends, as we have seen, solely upon the fre- quency of the repetition of the impulses ; the intensity, on their violence; and the quality, on the peculiar laws which regulate the molecular motions in any particu- lar instance. All sounds, whatever be their intensity or sounds having quality, in which the elementary impulses occur ^vithsame ritch>or ia x . unison. the same frequency, have to the ear the same pitch, and are said to be in unison. It is on the pitch alone that the whole doctrine of harmonies is founded. 8 91. The means by which a series of equidistant im- ° ^ x Musical sounds pulses can be produced mechanically upon the air are mechanically very various. If a toothed wheel be made to turn with Produced; a uniform motion while a steel or other spring is held against its circumference with a constant pressure, each tooth as it passes will receive an equal blow from the spring, and this being communicated to the air, a wave of sound will proceed from the place of collision. The 96 NATURAL PHILOSOPHY. The siren; A series of palisades; Whistling of a bullet, number of such blows in a second will be known when the angular velocity of the wheel and the number of teeth upon its circumference are known, and thus every pitch may be identified with the number of impulses which produce it. The Siren, another instrument by which the same results may be evolved, has been de- scribed in § 48. A series of broad palisades, placed edgewise in a line running from the ear, and equidistant from each other, will reflect the sound of a blow struck at the end nearest the auditor, producing a succession of echos which reach the ear at equal intervals of time, thus producing a musical note whose pitch will be determined by the number of reflexions in each second of time. This number will be equal to the quotient arising from dividing the velocity of sound by twice the distance between two adjacent palisades. A similar ac- count may be given of the singing sound produced by a bullet while moving through the air and turning rapidly about its centre of inertia. The angular motion of the bullet being uniform, the actual velocity of its surface on one side will be greater than that on the other, and any inequality in the figure of the bullet will be made to vary its action upon the air periodically, thus producing a musical sound. Most ordinary § 92. The most ordinary way of producing musical way of causing soun(js js ^0 set }n vibration elastic bodies, as stretched musical sounds; 7 strings and membranes, steel springs, bells, glass, co- lumns of air in pipes, &c, &c. All such vibrations con- sist in a regular alternate motion to and fro of the molecules of the vibrating body, and are performed in strictly equal portions of time. They are, therefore, adapted to produce musical sounds by communicating that regularly periodic initial impulse to the aerial mole- cules in contact with them, from which such sounds re- Modes suit. We proceed to consider their modes of production, and especially in the first and last named cases, these being the most simple. k ELEMENTS OF ACOUSTICS. 97 VIBPvATIONS OF MUSICAL STRINGS. §93. If a string or wire be stretched between two vibrations of fixed points, and then struck or drawn aside from itsmusicalstrin^' position of rest and suddenly abandoned, it will vibrate to and fro till its own rigidity and the resistance of the air bring it to rest; but if a fiddle bow be drawn across it, the vibrations will be renewed and may be maintained for any length of time, and a musical sound will be heard whose pitch will depend upon the greater or less ra- pidity of the vibrations. Thus, if 3INbe -. -, Fig- 52. any stretched cord, struck at right an- M --^ ^.s -^y ^.^ -_/---N" niustration; gles to its length at 0, it will be suddenly bent at that point into the curved or wTaved shape indicated by the dotted line S, which shape will run along the cord in both directions till it meets with some obstruction to its further progress, when it will be either wholly or partly reflected, and return upon its course in a manner and for the reasons to be explained wave runs along presently, the successive positions in the diverging mo- tnecordiab°th i or cvrr i «t i r* nt directions; tion being o , o , cvc, on the one side, and o,, S„, &c, on the other. § 9i. To find the velocity with which the wave runs to find the along the cord, it is plain that we may either regardvelocity of this -,-/•. wave motion; the cord as continuous, or as being composed of a series of detached points, kept in relative position by their mu- tual attractions for each other, each point being loaded with the mass of so much of the cord as extends half way on either side to the adjacent point, and of which the length is equal to the distance between any two con- secutive points. 98 NATURAL PHILOSOPHY. M- Fig. 53. "JL- JJ' 2) A a TV Suppose 31X to be the cord's position of illustration; rest, and the wave to proceed in the direc- tion from C to D; let the point A, be just on the eve of motion and the place B', the position of the point B at the' same instant. suppositions; While, therefore, the actual motion of the point B has been from B to B', that of disturbance has been from B to A. The duration of these simultaneous motions is indefi- nitely short; the motions themselves may, therefore, be regarded as uniform. Hence, denoting the actual velo- city of the point B by v, and the velocity of the dis- turbance by V, we have Consequences; V V::B B' : A B. v = V. BB' AB or, denoting the angle BAB' by

in which case, B B' AB = tan ' to the motive force; and therefore the velocity of A, in the small time ty which velocity will be equal to that of Bf wjien D begins to move, will be given by the relation G g l. t Velocity of a V = ---- ' tan ?' 'A t> ' particle in small w AH 1 . timet; But AB~ V * 100 NATURAL PHILOSOPHY. and denoting by Lt, half the length of the cord whose weight is equal to the tension C, we have Value of tension Q — 2 W. L C; ' which values substituted above give velocity of 2 g Ir tan 9 particle in small TT ' time t. and replacing tan

the fixed obstacle tend- ,„____________^~~~~^, ^ ing, when the pulse -* reaches the latter, to____________y ^ >-" n—ljvr move at right angles to the cord's length, ^ ^~n____________v_^ ijy will be resisted by the stationary molecules ; ^—^ ^ ^ ^-------------W the reaction will throw -* them to the opposite jf,------r—j---------------\tt K~-y Pulse thrown to side, and this reaction extending to the mole- cules behind, the pulse will pass to the oppoite side 0freturns; —> the opposite side extending to the mole- of the cord and |02 NATURAL PHILOSOPHY. Both ends of the the cord and return along its entire length, following after the direct pulse in the same direction. If the second end be fixed, the direct pulse proceeding towards it will conduct itself in the same way; the reflected pulses will proceed to meet each other, and being on the same side of the cord will conspire at their place of union to Reflected pulses produce a single resultant pulse, in which the molecules meet and of the cord will depart from their places of rest by the sum of the distances of the same molecules in the com- ponent pulses. These component pulses will, however, separate and are immediately separate, again reflected; and proceed towards the fixed ends, where J ! V_^ ^~-y ,2V" they will be reflected as before, and return to meet again, having once more changed sides. The point of Ml Meet a second second meeting will be time at the point at fae place of primitive disturbance, from which the of primitive # 7 disturbance; waves will depart, as before, to undergo the same round ; and thus, but for the resistance of the air and want of perfect elasticity in the cord, the latter would vibrate for ever. But every pulse communicated to the air, is an elimination from the cord of so much of its living force, and as this must soon become exhausted, the cord Cord brought to Will COUie to rest. M\—<*—---------------v, ^—\y M\-------------■•" v>-----------'.....^----[W / \ rest. § 96. Suppose the whole length of the Fig' 56' Whole length of corc} ]£ jsj to be de- 7r h l' cord divided into uUl----------------------—^------------LV two parts; noted by Z = l( + V, of which lt represents the distance 0 M, from the point of primitive disturb- ance 0, to the fixed obstacle on one side, and V the dis- tance 0 X to the obstacle on the opposite side. Then denoting the time of describing V by t', and that of de- scribing ld by td, we have ELEMENTS OF ACOUSTICS 103 V = V. t\ Lengths of the lt = V.tt; parts; whence ,/ L Times required T7~ ' for a pulse to pass along them. *, = ^-; ' y ' the first pulse being reflected at X, will describe the entire length V + l{ in the reverse direction in the time ,, I 7 Time in which v i vi — tt- "T" -r-r 5 the first pulse describes tho whole length; and the second pulse being reflected at M, will describe the entire length V + ld, in the same time, or "lr I The same for the a + / = .=. + second pulse; V "*" V ' the first pulse being reflected a second time at M, will describe the length V in the time 7 • Time in which t. —-— first pulse passes l — jr V over second part; and the second pulse being reflected a second time at N, will describe the distance V, in the time V That in which 0 =i , second pulse V passes over first part; and at the expiration of all these times the pulses will 104 N A TURAL PHILOSOPHY. Pulses return to starting point; Conspire and produce a resultant pulse: be at their first starting point, and each having been twice reflected, they will be on the same side of the cord that they were originally; they will, therefore, pro- duce a resultant pulse precisely the same, abating the qualification due to the air and imperfect elasticity, as that produced by the initial impulse. Hence, if T de- note the time of one complete vibration of the cord, that is to say, the interval between the instant of primitive disturbance and that at which the cord resumes its in- itial condition, we shall have, by taking the half sum of these several intervals—because both pulses are mov- ing during the same time, Time of vibration of the cord; T=2 (V + t) = 2(r + Z') = *Jk x i} y y and replacing V by its value, Equation (34), The same reduced. T = 2 L V2(/Zt (36) Example; Example. Taking the example of § 94, in which Z = 73, and 2 Z, = 1036,83 feet, we find* Result T 2. 73 V 32,18 . 1036,83 = 0,799. § 97. The truth of the foregoing theory has been fully confirmed by the experiments of Weber. Tie stretched a very uniform and flexible cotton thread fifty-one feet two inches in length, weighing 864 grains, horizontally, by a Experiments; known weight. The thread was struck at six inches from the end, and the time of the wave's running a certain number of times over the length of the string, back- ward and forward, carefully noted by means of a stop- watch that marked thirds (the sixtieth part of a second). ELEMENTS OF ACOUSTICS. 105 The mean of a great many trials, agreeing well with Mean of results; each other, gave the results in the following table : Tension in grains. Length run over by the wave. Time in thirds. Time of running over the length 102,33 in thirds by observation. Time by calcu-lation from the formula T-. 2i-V^gL, 10023 10023 10023 33292 69408 102*4 204,7 409,4 409,4 409,4 46 92 184 99 65 46 46 46 24,72 16,25 46,012 46,012 46,012 25,246 17,485 i Table. A more complete confirmation could not have been desired. The slight discrepancies are doubtless owing to a want of perfect uniformity in so long a thread, which must necessarily have formed a catenary of sen- sible curvature. Denote by X the number of vibrations performed in a given time Tt, then will T = I X Time of one vibration of the cord; which substituted for T in Equation (36) gives, after taking the reciprocal of both members, X V 2 g Ls I 21 (37). Reciprocal of the same. In the foregoing equations 2 Ld, denotes the length of the cord of which the weight measures the tension. Denote this weight by W, the diameter of the cord by Dy and its density by d; then will W= n. ^1 .2L;.d.g Weight of cord whose length measures the tension; 106 NATURAL PHILOSOPHY. whence Length of half this cord; L, = _*Z_ 7T.D2.d.g which substituted in Equations (36) and (3T), give Time of vibration; T= v* D.Z. vd (38) Its reciprocal; T. y/ir vw D.L.Vd' (39) Rule first; Eule second. from which it appears that, the time of vibration of a tense cord varies as its length, diameter and square rdot of its density, directly ; and the square root of the stretch- ing force, inversely. And that, the number of vibrations performed by a tense cord in a given time, varies as the square root of the stretching force directly, and the diame- ter, length and square root of the density inversely. vibrating cord g 93. ^he tense ends and struck cord MX being fixed in the middle; at ^fa en(Js an(J Jn a Fig. 57. Mv _Z -^1 -ij\r Primitive pulse resolved into two; state of vibration, ap- ply the finger, or any other partially obstructing cause, at the middle point F, and then withdraw it. The law of Equation (34), will be suddenly interrupted at this point, the progressing pulse will be resolved into two, one of which will continue to move in the same direc- tion and on the same side of the cord, while the other will be reflected and return along the opposite side. These component pulses having equal distances to tra- vel before they reach the ends, will be reflected at the fixed points at the same instant, return on opposite sides of the cord, and meet in the centre. They will, ELEMENTS OF ACOUSTICS. 107 Fig. 53. M>- 0 MT -^ therefore, solicit simul- taneously the central point 0, in opposite di- rections, and if the pulses be equal, they will wholly interfere at that point, which must, therefore, remain stationary. The effect of the reciprocal action of the two waves being to fix the point 0, these waves will both be totally reflected there, will return to the ends, be again reflected, return to the centre, from which they will be thrown back towards the ends, and so on till the living force of the cord is totally expended upon the air. Thus the two portions M 0 and 0 X of the cord may vibrate as though the point 0 had been originally fixed. If the finger be ap- The two reflected pulses interfere at the middle of the cord; Are thence again reflected and so on. Fig, 59. Jf\ 0 ~\JV My o -iJir Mv O' -ixr plied but for an in- stant at 0, at a dis- tance from 31 equal to one-third of the whole length MX, wdiile the wave is pro- gressing from 31 to- wards 0, the latter will be resolved, as before, into two component waves, one of which will continue to move towards X on the same side of the cord, the other will return to M on the opposite side. The distance MO being equal to one- half of 0 X, the return component will be wholly re- flected and change sides at M, and come back to the point 0 by the time the direct component arrives at X, where the latter will be totally reflected and pass to the opposite side of the cord. The component waves being now on opposite sides of the cord, and moving towards each other, one starting from 0 and the other from X, will meet at 0', half way from 0 to X, making X 0' equal to one-third of M X. Here they will interfere, be totally reflected, and proceed from Finger applied at one third the whole length from the end; Primitive pulse resolved into two; Component pulses interfere; 108 NATURAL PHILOSOPHY. Are again reflected; Again meet at their starting point and so on. Finger kept on the cord; One reflected component resolved into two; Reciprocal action between these two sets of components; Cord broken up into portions, each one vibrating. Fig, 59. M\ 0 ajt My Mh O .....^ ;' vibration; and in Equation (37), the number of vibrations in the time Td, X n V 2 q . I Number in time 2] 2 1 and for the number in one second, by making Tt equal to one second, -rr n y/ 2 g . I{ /i-. v Number in one ■^ == Q~~r .....V^1)* second. All of this is confirmed by experience. If the string Above of a violin, or violincello, while maintained in vibration deduction8 it-it ill i confirmed by by the action ot the bow, be lightly touched by the experience; fino-er, or a feather, exactly in the middle or at one-third of its length, from either end, it will not cease to vibrate, but its vibrations will be diminished in extent and increased in frequency, and a note will become audible, more faint but more acute than the original, or fundamental note, as it is called, and corresponding, in the former case, to 110 NATURAL PHILOSOPHY. illustrated by the & double, and in the latter, to a triple rapidity of vibra- violin; tion# q,|ie no|-e beard in the first case being, in the scale of musical intervals, an eighth or octave, and in the second a twelfth, above the fundamental tone. If a small piece of paper cut in the form of an inverted V, be set astride on the string, it will be violently agitated or thrown off if placed on the middle of a ventral seg- ment, but at the node will ride quietly as though the Harmonics. string were at rest. The sounds thus produced are termed Harmonics. Coexistence and superposition of small motions; Its application illustrated; Explanation; Results confirmed by experience. Fig. 61. § 100.-But further, according to the principle of the coexistence and superposition of small motions, referred to in § 56, any number of the various modes of vibra- tion of which a cord is susceptible, may be going on simultaneously. Thus, if we suppose a mode of vibration repre- sented by figure (a), in which there is no node, and another of the same cord represented by figure (b), with one node, to be going on at the same time, there will be a resultant vibration represented by the curve in figure (c), of which the ordinates are equal to the algebraic sum of the corresponding ordinates of the curves in figures (a) and (b). If a third mode of vibration, represented by figure (d), be superposed upon the other two, there will arise a resultant vibration represented by the curve in figure (e), of which the ordinates will be equal to the algebraic sum of the corresponding ordinates in figures (a), (b) and (d), or, which is the same thing, the algebraic sum of the corresponding ordinates of figures (c) and (d). This is also confirmed by experience. It was long known to musicians, that besides the fundamental note ELEMENTS OF ACOUSTICS. Ill of a string, an experienced ear could detect in its sound, Harmonic when in motion, especially when very lightly touchedsounds; in certain points, other notes related to the fundamental one by fixed laws of harmony, and which are therefore called harmonic sounds. They are the very sounds that may be heard by the production of distinct nodes as ex- plained in § 99, and thus insulated as it were from the fundamental and other coexisting sounds. § 101. The 3Ionochord is an instrument adapted to ex-Themonochord; hibit these and other phenomena of vibrating strings. It consists of a single string of catgut or metallic wire stretched over two fixed and well defined edsces to- wards its extremities, which effectually terminate its vibrations in the direction of its length; one end is permanently fixed, and to the other is attached a weight which determines the tension. The interval be-Essential parts; tween the two edges is graduated into aliquot parts, and the instrument is provided with a movable bridge or piece of wood capable of being placed at any point of the graduated scale, and abutting firmly against the string so as to stop its vibrations, and divide it into two equal or unequal parts, as the case may be. By the aid of this instrument may readily be found its use; the number of vibrations which corresponds to any given note of any particular instrument, as a piano-forte, for in- stance. To this end, it will only be necessary to know, wThen the note of the monochord is the same as that of the instrument, the distance I between the edges, the stretching weight, and the wreight of a unit's length of the string. The quotient obtained by dividing the former of these weights by the latter will give the va- lue of 2 Lt, in Equation (37), and making Tt equal to one second in that Equation, we have for the solution of the question X ■ ^ ^ ^ ' (42")Practical O 7" ^ ' formula. 112 NATURAL PHILOSOPHY. Number of This gives the number of impulses made upon the ear rrrelpTudingto *n a second, corresponding to the fundamental note. To any higher note, obtain the number whicli answers to any note sharper, higher, or more acute, we have but to apply the bridge and slide it to some position such that the portion of the cord between it and one of the edges gives the note in question; the scale will make known I, whicli in Equation (42), will give the number X. Harmonic tones § 102. The contact of a stretched cord wTith solid sub- pro uce y gtances is not the only means of producing its fundamen- causing air in J JL o *v*muvu motion to tal and harmonic tones. The sonorous pulses proceed- 3cXuord!tain§ fr°m a Crating cord are but the consequences of repeated conflicts between the elastic force of the cord and that of the air. The former impresses upon the air a certain amount of living force, and the latter by its reaction transmits this living force through the atmos- phere to a distance. Reverse the process. Impress upon the air the same motion, and subject a stretched cord to its influence. Action and reaction only change names, and the cord must take up the motion of the air. Two Two -ords near cords equally stretched, and in all other respects similar, Ih^^niadeto1^111 the length of one only a half, a third, or any aii- ww.te; quot part of the other, being placed side by side, and the shorter put in motion, the longer will soon assume a mode of vibration by which it will be divided into ventral segments, each equal to the length of the shorter cord. The sonorous pulses diverging from the shorter cord will arrive at the longer; and the mole- cules in the first of these pulses will, in their forward movement, press upon the stationary cord and give it a slight motion in their own direction. On the retreat of these molecules, the excess of aerial condensation will change to the opposite side of the cord; the latter will yield to the action of this inverted force and that of its its vibrations own elasticity, and pass to some position on the oppo- fcansforredtothe site side of its Place of rest, where being met by a second longer cord; onward pulse, it will be thrown back in the direction of ELEMENTS OF ACOUSTICS. Hg Synchronal its first motion, and thus made to undergo the same round as before. This process being repeated a number of times, the , cord will be set in full and audible vibration. But vibrations; these vibrations will obviously be synchronal with the aerial pulsations, and therefore, with the vibrations of the shorter cord, a condition that can only be fulfilled by d *J Longer cord is m the longer cord breaking up, as it were, into portions etfect broken up; of which the lengths are equal to the length of the shorter cord; for, the tension, diameter and density, of the cords being the same, the times can only be equal, Equation (38), when the vibrating lengths are equal. All motions of the longer cord which are inconsistent with this, though they may be excited for the moment by one pulsation, will be extinguished by the subsequent one. Hence, if two cords can have any mode of vibra- tion in common, that mode may be excited in either of them, and that only, by exciting it in the other. For example, if two cords, in all other respects alike, have lengths which are to each other in the proportion of 2 to 3, and if either be set in motion, the mode of illustration. vibration corresponding to a division of the first into two and of the second into three ventral segments, will, if it exist in the one, be communicated by sympathy to the other. Indeed, if it do not originally exist, it will, after awhile establish itself; for, all the circumstances whicli may favor such a division, however minute, will have their effect preserved and continually accumulated, and thus become sensible. And it is important to remark that whether the primi- tive portion disturbed be large or small, whether it occu- py the whole string at once or run along it like a bulge ; whether it be a single curve, or composed of several ven- tral segments with intervening nodal points, we must not forget that the motion of a string with fixed ends is no strin° withfixed other than an undulation or pulse continually doubled backends is analogous upon itself, and retained within the limits of the cord retainedwitMn instead of running off both ways to infinitv. certain limits. 8 114 NATURAL PHILOSOPHY. vibrations § 103. It is very seldom that the vibrations of a string seldom confined can ylQ jR faQ game p]ane. TllCV HlOSt COmillOllly COllsist to the same t i i plane; of rotations more or less complicated, except when pro- duced by the sawing of a bow across the string. The actual orbit described by any one molecule may be made matter of ocular inspection by throwing the solar rays througli a narrow slit so as to form a thin sheet of light. orbits described A polished wire stretched in such manner as to penetrate by particles may this siieet at right angles, will appear, when stationary, as be observed; ^ t^g^t spot where it pierces the light, but when in mo- tion, the point of intersection will form a continued lumi- nous orbit, just as a live coal whirled round appears like a circle of fire. The figures exhibit specimens of such orbits observed by Dr. Young. Fi?. 62. Specimens. VIBRATING COLUMN OF AIR OF DEFINITE EXTENT. Vibrating column of air of definite extent; § 104. The circumstances of the molecular vibrations of a stretched cord of indefinite extent, are, as we have seen, similar to those of a sounding column of air; and the facts which have been stated respecting a vibrating cord are equally true of a vibrating column of air of definite extent. ELEMENTS OF ACOUSTICS. 115 Fig. 63. B C B Thus, if such a cylindrical column be enclosed in a pipe A B — I, stopped at both ends by immovable stop- pers, and an impulse be com- municated in the direction C A, to one of its sections C, at the distance AC—I, from the end A, and B C=l', from the end B, this impulse will, § 69, give rise to two pulses running in opposite directions. In the pulse from G to A the air will be con- densed, and in that from C to B it will be rarefied. These pulses will be reflected at the stoppers, and the condensed pulse, after passing over the distance I be- fore and I' after reflexion, will meet the rarefied pulse at the distance I from the end B, and produce a com- pound agitation in the section C similar to that of the original disturbance ; thence the partial pulses will sepa- rate, and after each undergoing another reflexion will unite in their original point of departure, constituting, as it were, a repetition of the first impulse, and so on, till the pulses are destroyed by the gradual transmis- sion of the whole of their living forces through the sub- stance of the tube to the open air. If the section first set in motion be maintained in a state of vibration synchronous with the return of the reflected pulses, it will unite with and reinforce them at every return, and the result will be a clear and strong musical sound, resulting from the exact combination of the original periodic impulse with its echos. Tube closed at both ends, containing air; Impulse communicated to a section in direction of the length of the tube; Two pulses will be started running in opposite directions; Pulses gradually destroyed. Consequence of maintaining in vibration the section first disturbed. A Fte. 64 § 105. Let us suppose the section first set in motion and so maintained, to be exactly in the middle of the pipe. Then, when once the pe- riodic pulsation of the contained air is established, the motion will consist of a constant and regular fluctu- JS Middle section maintained in vibration; 116 NATURAL PHILOSOPHY. Air condensed in one half and rarefied in the other; Positions of greatest and least condensations and rarefactions; Several columns end to end; Illustration; Nodes; Ventral segments; Distance between two alternate nodes. ation to and fro of the whole mass, the air being always condensed within one-half of the pipe while it is rare- fied in the other. The greatest excursions from their places of rest will be made by the molecules in the middle, while the molecules at the ends abutting against the solid stoppers will have the least motion, the ex- cursion made by each intermediate molecule being greater in proportioii as it is nearer the centre. On the other hand, the rarefactions and condensations are greatest at the extremities and diminish as we approach the middle, where thej are the least. Now, conceive several such columns of vibrating air to be equal and to be placed end to end, so that the condensed portions shall be turned towards each other; it is plain that all the stoppers, except the ex- treme ones, may be removed without in anywise sen- sibly changing the interior motions, and there will re- sult a single column of air broken up into equal por- tions vibrating in a manner similar to that of the ventral segments of a tense cord, «- -* <- § (J8, the nodes being at X and Y, where there will be alternately a maximum and minimum of condensation, the bellies lying between — in the middle of which the condensation will be the least. It is also obvious that the distance XX, be- tween two alternate nodes, will be the shortest dis- tance from any one section of air to another having the same phase, and that this distance answers to the length of a wave of the same pitch propagated in an indefinite column of air. Fig. 65. Y X Y An opening in the middle of a segment; § 106. At C, half way be- tween two consecutive nodes, or in the middle of one of the cylinders A B, let an opening be made; and sup- x Fig. 66. ELEMENTS OF ACOUSTICS. 117 pose a vibrating body to be inserted whose vibrations And a vibrating are executed in equal times with those in which thebodyiutroduced; excursions to and fro of the included aerial sections are performed in the stopped pipe. Its vibrations will be communicated to those of the contained air, the latter will be maintained and strengthened, and the sound from the pipe will become full and clear. Such Embouchure. an aperture is called an embouchure. Next conceive one-half one half of the B C, of the cylinder A B, Fig- 6T- !ylimkhr 7,acc ELEMENTS OF ACOUSTICS. 121 musical sound. The column of air vibrates in the mode Node in the . middle of the represented in the figure, in which there is a node in entire segment, the middle, and each ventral segment is only half a complete one. §112. To find the time of vibration or the number PiPe °Pen at of vibrations in a given time both ends' fe Fig. 71. $<- corresponding to any mode of - vibration, denote by m the number of nodes in a pipe open at both ends; the number of complete ventral segments between them will be ., Number of m — 1; complete ventral segments; and denoting the length of the pipe in feet by Iu, the length of each- complete ventral segment will be T Length of each ---j segment; m and denoting the velocity of sound by V, and the time required for the sonorous pulse to traverse one seg- ment by T, we shall have Time of 1 T J7— __ . ".......(43) describing one r ofsound v= ,v . *-» v " y"'.... (50) 2m-l Z in a gas: and making m and n each unity, T \/ O /y T, , Same reduced; F = LLi^Li^^i.....(51) which furnishes a readv means of finding the velocity of sound in any gas or vapor. For this purpose, fill a pipe of known length with the gas in question, and set it to vibrating by any proper means, so as to call forth its fundamental tone. Adjust the bridges of a 31ono- Practical use of p. . in ^ne aD°ve chord so that the fundamental tone oi its string shall forraula have to the ear the same pitch; measure the length of the string between the bridges and substitute this length for I in Equation (51), and the velocity sought becomes known. It was by this method that Chladni, Vaknees, Frameyer and Moll ascertained the velocity of sound in various media. For a detailed account of the structure and manage-Account of™,es- reeds, &c. ment of the embouchures of pipes, and a vast amount of interesting matter on the subject of reeds, &c, &c, the reader is referred to Sir John Hersciiel's most va- luable Monograph of Sound, articles 197 to 207, inclu- sive, as published in Vol. IV. of the Encyclopedia Metropolitana. 126 NATURAL PHILOSOPHY. VIBRATIONS OF ELASTIC BARS. vibrations of § 117. Bars of a cylindrical or prismatic shape are bars; susceptible of sonorous vibrations as well as cords, and columns of air. But as such bodies are nearly equally elastic in all directions, transversely as well as longitu- dinally, their vibrations do not obey the same laws as Transversal and those of strings. Transversal vibrations may be excited longitudinal; ^ g^^ing a ^ar cr0sswise, and longitudinal vibrations by striking it in the direction of its length. Laws governing In bars made of the same substance, the acuteness the pitch in 0f the pitch in transversal vibrations is directly as the Xrations; thickness, and inversely as the square of the length of the bar. In bars made of different substances, it is found that the degree of the body's elasticity greatly influences the character of the pitch; thus steel gives a higher pitch than brass. To produce these vibrations, the bar may be either secured at both ends, or its ends may be made merely to rest on Fis- 74« some fixed objects; or one end v^m^^^^Mm^m^^M^ may be fastened while the other is Means of producing these free, or lastly, both ends may vibrations in ^ free^ the rods being support- ed at two points. We have an illustration of these kinds of vibrations in the jews-harp, musical boxes, &c. § 118. When a "bar is struck upon one of its ends in the direction of its length, the blow will give rise to a condensed pulse, which will proceed towards the other vibrations in end like that of a column of air. It will be reflected bars; back and forth alternately at the two ends, according to the principles of §106 and §107, and this will con- tinue till its living force is wholly transmitted to the bars. mi—,_____., v ■■■'.:■.■.■ ■■.,..-..,,_ Longitudinal ELEMENTS OF ACOUSTICS. 127 air and wasted in space. If the rod be of glass, the solid rods give sound emitted will be extremely acilte unless its length sounds than be very great; much more so than in the case of a columns of air; column of air of the same length. The reason of this is, the greater velocity with which sound is propagated in solids than in air. When the bar is short the re- flexions at the ends, which determine the successive impulses upon the air and therefore the pitch, succeed each other with great rapidity. The velocity in cast iron, for example, being 10^ times that in air, a rod Glass>steel> &c* of this metal will yield a fundamental sound when lon- gitudinally excited, identical with that of an organ-pipe of Ti elastic rods admit of a third, viz., that by rotation. It is most easily generated in cylindrical bodies, by secur- ing one end in a vice, and communicating to the other a rotatory motion by means of a bow or by friction. An alternate expansion and contraction ensue in a direction perpendicular to its axis. Different high and low notes succeed each other, of which, as yet, no use has been made in music. OF THE VIBRATIONS OF ELASTIC PLATES AND BELLS. Vibrations produced in plates; If elastic plates, of glass or metal in particular, be held tightly either by the fingers or by means of a clamp, at any one point, and the bow of a violin be drawn across the edge of the plate, sonorous undulations are imme- diately produced. These oscillations resemble those of elastic rods, inas- ELEMENTS OF ACOUSTICS. 129 much as the surface is divided into a greater or less num- ber of perfectly symmetrical parts, and such as are con- terminous, vibrate in opposite directions. The boundary lines of these several parts are all in a chiadni's state of repose, and form nodal lines; their position de-sonorous figlires; pends on the places at whicli the plate is held and ex- cited, as one of these nodal lines invariably runs througli the point at which the plate is held, whilst the plate itself receives the vibratory motion at the other point. These lines form certain peculiar figures, called, after their dis- coverer, Chladni's Sonorous Figures. To make these figures visible, and to render them per- >r , ,. ° J r Means of making nianent, strew some light sand or dust over the plate; these visible; they may also be seen if a small quantity of water be poured on the plate, nay, even by the rays of light falling on it. Wiieatstone remarks that, in using the last-named mode, still more delicate divisions in the figures were observable. These sonorous figures are composed sometimes of Their shapes right lines, sometimes of curves either parallel to or^^thT intersecting each other. The shape of the ^plate greatly p^te; affects them, as they are differently arranged, accord- ing as it may be a square, a rectangle, a triangle, a circle, an ellipse, or some other figure. A perfectly dis- tinct and well-defined figure is produced only when the plate gives a very clear sound. By experiments made on such plates the following Laws; laws were detected by Culadni : 1. Any particular pitch will always produce the same figure with the same plate; but a small change may often be produced in the figure by slightly changing the place at which the plate is held without causing any difference in the pitch. If the pitch be changed, First law; the existing figure disappears at once, and a new one arranges itself. 2. The gravest pitch any plate gives is accompanied by the simplest figure, and the higher the pitch the more second law; complex the figure, i. e. the more nodal lines there will be. 130 NATURAL PHILOSOPHY. Third law. Experimental illustration; 3. If similar plates of various sizes be treated in the same manner, similar figures will be generated in each; by the same treatment, we mean that they shall be held at the same point, and that the bow shall pass over corresponding points in each. The pitches will, however, differ, for the larger plate will give out the oraver sound; and if their dimensions be equal, the stronger will give the acuter pitch. § 120. If the plates be strewed with fine sand, and held at the point a, whilst the bow be made to pass at b, the figures here depicted will in each case be produced. Fig. 76. a-'i ft t. a III u.. "iMl'l' :■* H a .-§ ■>''' $ A striking effect is obtained by making the same figure on several plates of equal size and similar form, and then so arranging them as to make one figure on union of several a larger scale. The figure thus produced will be both a compound and connected one, and such as may not unfrequently be met with on a large plate. If a large square be formed out of four squares, bear- ing the figures I. and II., we shall have the following: plates of equal size; Fig. 77. Fig. 78. The effects. ELExMENTS OF ACOUSTICS. 131 \ If the large square plates be held at a, touched at a', Particular case; and a bow be drawn across at J,'similar compound fig- ures will be generated. Cymbals, the Chinese Tam-tam or Gong, &c, are prac- Examples; tical applications of sonorous plates. § 121. The vibrating motions of sonorous bells resem- sonorous beiis; ble those of circular plates. In this case, too, the most acute pitch is accompanied by the most complex figure. To render these vibrations visible, fill a bell-shaped glass father more than half full of water; draw a vio- lin bow across the rim, and at the same time touch the glass at two opposite points of the rim with the fingers. The surface of the water will acquire an undulatorv mo-Means of making tion, and to make the sonorous figures permanent, strew lhefce acou! !c 1 or? figures visible; the surface of the water with any light and exceedingly fine powder, as semen lycopodii. If the point excited by the bow be at a distance of 45° from that touched by the finger, a four rayed star marked III. of the last article will result; but if the distance be 30°, 60°, or 90°, the six-rayed star, marked IV., w^ill appear. Such a cup gives musical sounds when rubbed with the The moistened finger. The vibrations of the glass in this musical sounds. case result from torsion, and this is the principle of the well known finger glass. COMMUNICATION OF VIBRATIONS. § 122. The numerous experiments of M. Savart abun- communication clantly show that the molecular motions of one body areof molecular communicated to another, when there exist between them any intervening media, and this the more effectually as the connection is the more perfect. But not only this; they also show that the molecules of the neighboring bodies are agitated by motions both similar in period and parallel in direction to those of the original source of mo- its peculiarity; 132 NATURAL PHILOSOPHY. tion. Of these experiments we have only room for such as have a direct bearing upon the nature and structure of our organs of hearing. § 123. Take a thin membrane, moistened tissue paper will answer every purpose, and stretch it over the mouth Experimental 0f a common bowl or finger glass, place it in a horizontal position and strew fine sand over its surface. Hold a glass plate, covered with fine sand or dust, horizontally and di- rectlv over the membrane, and set it in vibration so as to form Chladni's acoustic figures; these figures will be im- mediately and exactly imitated in the sand on tbe mem- Effect when the brane, and this will be the case to whatever lateral posi- giassand tjon wjthin the sphere of sufficient action to move the par- pwawei"1 tides of sand, the plate may be shifted, provided it retain its parallelism to the membrane. Effect when § 124. But instead of shifting the plate laterally, let its inclined to one plane be inclined to the horizon. The figures on the membrane will change though the vibrations of the plate remain unaltered, and the change will be greater, the greater the inclination of the plane of the plate. And when it becomes perpendicular to the horizon and there- fore to the surface of the membrane, tbe figures on the latter will be transformed into a system of straight lines when parallel to the common intersection of the two planes; perpendicular to an(j ^e particles of sand, instead of dancing up and down, will creep in opposite directions to meet on these lines. One of these lines always passes through the centre, and the whole system is analogous to what would be produced by attaching a cord to the centre of the plate, and, having stretched it very obliquely, setting it in vibration when shifted by a bow drawn parallel to the surface. In a word, the lateral^; vibrations of the membrane are now parallel to its sur- face, and they preserve this character unchanged, how- ever the plate be shifted laterally, provided its plane when the plat© is be kept vertical. If the plate be made to revolve about revolved about .^ vert}ca| diameter, the nodal lines on the membrane its vertical ' diameter. will rotate, following exactly the motions of the plate. ELEMENTS OF ACOUSTICS. 133 §125. Nothing can be more decisive or instructive inference; than*this experiment. It shows us that the motions of the aerial molecules in every part of the spherical wave propagated from a vibrating centre, instead of diverging like radii in all directions, so as to be always perpen- Principle of dicular to the wave surface, may be parallel to each tr^nsyersal ' * a vibration; other and to the wave surface. The same holds good in liquids also. § 126. So long as the sound of the plate, its mode of vibration, its inclination to the plane of the membrane, and the tension of the membrane continue unchanged, the nodal figure on the membrane will continue the Conditions to same ; but if either of these be varied, the membrane lnsule 1 7 permanence or will not cease to vibrate, but the figure on it will be figure; changed accordingly. Let us consider separately the effects of these changes. § 127. All other things remaining the same, let the pitch of the sounding plate be altered, either by loading Difference it or changing its size. The membrane will still vibrate, w?en a o o i membrane and differing in this respect from a rigid lamina,, which can rigid lamina; only vibrate by sympathy with sounds corresponding to its own subdivisions. The membrane will vibrate in sym- pathy with any sound, but every particular sound will be accompanied by its own particular nodal figure, and as the pitch varies, the figure will vary. Thus, if a slow air be played on a flute near the membrane, each note inustration. will call up its particular form, which the next will efface to establish its own. § 12S. Next suppose the figure of the plate so to vary a f fl as to change its nodal figures; those on the membrane of the piate; will 'also vary ; and if the same note be produced by dif- ferent subdivisions of different sized plates, the nodal figures on the membrane will also be different. 134 NATURAL PHILOSOPHY. Effect of change of tension; Effect of moisture; § 129. If the tension of the membrane be varied ever so little, material changes will take place in its nodal figures. Hygrometric variations are sufficient to produce these changes. Indeed, the fluctuations arising from tin's cause were so troublesome in the case of tissue paper, that it became necessary to coat the upper surface with a thin film of varnish. By far the best substance for ex- hibiting the results of these beautiful experiments is var- nished paper. Moisture diminishes the cohesion of the fibres, and renders them nearly independent of each other, and sensible alike to all impulses. § 130. Between the nodal lines formed by the coarser particles of sand, others are occasionally observed, formed only of the finest dust of microscopic dimensions. This secondary nodal is a most important fact, as it goes to show that diffe- lines; rent and higher modes of subdivision coexist with the more inference: elementary divisions which produce the principal figures. The more minute particles are proportionally more re- sisted by the air than the coarser ones, and are thus Explanation; prevented from making those great leaps which throw the coarser ones into their nodal arrangement. They rise and fall with the greater divisions of the surface, and are only affected by those minute waves whicli have a smaller amplitude of excursion and occur more frequently, and form their figures as though the others did not exist. These secondary figures often ap- pear as concentric rings between the primary ones, and frequently the centre of the whole system is occupied as a nodal point. Sensibility of some membranes; Exploring membranes § 131. So sensitive are some varieties of stretched mem- brane to the influence of molecular motion that they have been employed with success in detecting the ex- istence and exploring the extent and limits of the most delicate, continuous and oppositely vibrating portions of air. When so employed they are called exploring mem- ELEMENTS OF ACOUSTICS. 135 branes. The most highly interesting application of the Application of properties of stretched membrane is in the " membrana the Prorerties of stretched tympani^ Of the ear. membrane; Fig. 79. Illustrations; .C?v, Illustrations; THE EAR. § 132. The auditory apparatus, called the ear, is a E^ential art8 collection of canals, chambers, and tense membranes, of the ear; whose office is to collect and convey to the seat of hearing, the vibrations impressed upon the air by sono- rous bodies. Beginning on the exterior and proceeding inwards, 136 NATURAL PHILOSOPHY. Wing; Auditory duct; Cavity of the drum; Fig. 80. Labyrinth; Vestibule; Fenestra ovalis and conchlea; we find a cartilaginous funnel A A, called the wing; a canal b b, called the auditory duct, leading to an interior chamber B B, called the cavity of the drum ; and behind this a system of canals of con- siderable complexity, call- ed the labyrinth, consist- ing of three semi-circular tubular arches m, m, m, originating and terminat- ed o ing in a common hall n, called the vestibule, which communicates with the cavity of the drum by a small opening I, called the fenestra ovalis, and is prolonged in the opposite direction into a spiral cavity o, called the conchlea. The auditory duct is closed at its junction with the cavity of the drum by a tense Drumof the ear; membrane r, called the drum of the ear, as is also the fenestra ovalis by a similar membrane. The whole ca- vity of the labyrinth is filled wTith a liquid in which are Auditory nerve; immersed the branches of the auditory nerve, wherein is supposed to reside the immediate seat of the first im- pression of sound. "Within the cavity of the drum are four small bones united by articulations so as to form a continuous chain; the first f is called the hammer, the second g, the anvil, the third i, the ball, (os orbicularis), and the fourth k, the stirrup, from the resemblance which its shape bears to that of the common stirrup. The han- dle of the hammer is attached to the drum, and the stirrup to the membrane which closes the fenestra ova- lis ; and thus the aerial vibrations, first collected by the funnel-shaped wing of the ear, and transmitted through the auditory duct to the drum, are conducted onwards by the articulated bones to the auditory nerve in the labyrinth, which receives them at the window of the vestibule. The cavity of the drum is connected with that Eustachian tube; of the mouth by a canal d, called the Eustachian tube, Hammer, anvil, ball and stirrup; ELEMENTS OF ACOUSTICS. 137 which serves to keep the cavity of the drum filled with its use. air of uniform density and temperature; a condition which appears to be necessary in order that ,the different parts may perform their functions with accuracy. If this be stopped, deafness is said to ensue, but as Dr. Wollaston has shown, only to sounds within certain limits of pitch. If the membrane which closes the labyrinth be pierced Deafties8 and its fluid let out, complete and irremediable deaf- produced. ness ensues. From some experiments of M. Flourens on the ears of birds, it appears that the nerves en- closed in the several arched canals of the labyrinth have other uses besides serving as organs of hearing, and are other uses of the instrumental, in some mysterious way, in giving animals ne*vesof the the faculty of balancing themselves on their feet and directing their motions. ear. MUSIC, CHORDS, INTERVALS, HARMONY, SCALE AND TEMPERAMENT. § 133. Our impression of the pitch of a musical sound th^affectour depends, as we have seen, entirely upon the number impression of a of its vibrations in a given time. Two sounds whosemusica " vibrations are performed with equal rapidity, whatever be their difference in intensity and quality, affect us with the sentiment, of accordance, which we call uni- soii, and impress us with the idea that they are simi- lar. This we express by saying that their pitch is the same, or that they are the same note. The impulsesEffectoftwo J # ■*■ sounds in unison; which they send to the ear through the medium of the air, occurring with equal frequency, blend and form a compound impulse, differing in quality and intensity from either of its components, but not in the frequency of its recurrence, and we judge of it as of a single note of intermediate quality only. oftwonotin But when two notes not in unison are sounded to-unison: 10 138 NATURAL PHILOSOPHY. Concord or discord. gether, most persons distinctly perceive both, and can separate them in idea, and attend to one without the other. But besides this, the mind receives an impres- sion from them jointly whicli it does not receive from either when sounded singly even in close succession ; an impression of concord or of discord, as the case may be, and hence the mind is pleased with some combinations, displeased with others, and it even regards many as harsh and grating. Harmony, chord, melody. Music. Concordant Bounds; Discordant sounds ; Limit of simplicity in music. § 134. The union of simultaneous and concordant sounds, is called Harmony. Every group of simultaneous and concordant sounds, is called a Chord in harmony. A succession of single sounds makes 31elody. To discover and discuss the laws of harmony and melody, is the object of musical science ; to apply these laws to the production of certain effects in musical composition, is the object of musical art. Science and art, thus combined, constitute that department of knowledge properly called 31usic. Now it is invariably found that the concordant sounds are those, and those only, in which the number of vibra- tions in the same time are in some simple ratio to each other, as 1 to 2, 1 to 3, 1 to 1, 2 to 3, &c, and that the concord is more pleasing the lower the terms of the ratio are and the less they differ from each other. While, on the other hand, such notes as arise from vibra- tions which bear no simple ratio to each other, as 8 to 15, for instance, produce, when sounded together, a sense of discordT and are unpleasant. By the constitution of the ear, ratios in which 7 and the higher primes occur are not agreeable; why, cannot be told, but simplicity must end somewhere, and in music this seems to be about the point. This is the natural foundation of all harmony. § 135. The relative effect of any two sounds is found to be always the same as that of any other two in which the ratio of the vibrations is the same. Thus sounds of ELEMENTS OF ACOUSTICS. 139 which the vibrations are respectively 12 and 18, produce compound the same effect as those whose vibrations are 40 and 60, produced for same effect; 18 60 3 12 40 2 ' and we say that according as the first and second sounded together, are pleasant or unpleasant, so are the third and fourth; also, if an air beginning on the first sound re- quire an immediate transition to the second, then, the same air beginning on the third will require an immediate transition to the fourth. § 136. The relative pitch of two sounds is called an intervai; interval. Its numerical value is expressed in terms of the graver sound, represented by the number of its vibra- tions in a given time, taken as unity. The value ofItsnnmei.ical an interval is, therefore, always found by dividing the value found; number of vibrations of the acuter note in a given time by the number of vibrations 'of the graver note in the same time; thus, the interval of two sounds, one of which is produced by two and the other by three vibrations in the same time, has for its measure f. If 18, 23, and 30, be the numbers of vibrations of three sounds Examples; in the same time, and we wish to find a fourth sound which shall be as much above the third as the second is above the first, we say, 18 : 23 :: 30 : x = 3Q ' 23 = 38|. 18 § 137. Next to unison, wherein the vibrations of the two sounds are in the ratio of 1 to 1, the most satisfactory vibrations in the , . , . i • i ,i «i i* . .1 .. ratio of 1 to 2; concord is that m which the vibrations are in the ratio of 1 to 2. The effect of this is not only pleasing, but it always gives rise to the idea of sameness; insomuch that if two instruments were made to play together in such manner that the sounds of the one should always 140 NATURAL PHILOSOPHY. Ghe the be of twice as many vibrations as tbe simultaneous sounds imi^ion of of th Qth t} ^ be univer6ally admitted to be different shades J J of the same air; playing the same air, with only that sort of difference which is heard when a man and a boy sing the same tune together. Now, take a tense string, and call the sound emitted Two tense strings 1 °7 whose vibrations from it C, and, for the reasons given above, let the sound from a string of double the number of vibrations be called C. Let us seek for the simplest fractions which lie between 1 and 2, up to the prime 7, and we shall find, are in the ratio of 1 to 2; 1 i. i. 5. 25 3? 35 45 1 1 2? 55 series of and these, arranged in the order of magnitude, give, vibrations that n. -, . _, 1 r\ ji j_ will produce ^ter placing 1 and 2 on the extremes, agreeable musical sounds. 1, j, £, 1, f, 5, 2. A set of tense strings, or of pipes, so arranged that the first makes one vibration while the second makes f of a vibration, the third f of a vibration, and so on to the last, which makes 2, wrill emit sounds every one of which will be agreeable when sounded with the first. sounds which § 138. But it is found that the frequent repetition of are pleasing and soun(js which are very near to each other is not pleasing to those which are . , . . r not so. an uncultivated ear, and that the frequent repetition ot sounds too far from each other is not pleasing to the ear after a little cultivation. Taking the intervals of the above series, we find that from the Examination of the above series; 1st to 2d is | 2d to 3d is 3d to 4th is 4th to 5th is f 5th to 6th is f 6th to 7th is 2 -1 = 4 6 • 5 5 25 2 4 11 1 5 1 • 8 5 l_P 9 6 . 6 5 ELEMENTS OF ACOUSTICS. 141 The interval between the 1st and 2d, and that between Defects in this the 6th and 7th are too great, while the interval betweens< the 2d and 3d is too small for frequent repetition. A new sound must, therefore, be substituted for the second one of the scale, and of such value as to increase the interval between the second and third and diminish that Additions . _ required; between the first and second, while an additional sound must be interpolated between the 6th and 7th. Denote the first of these by x and the second by y, then will the series of ratios stand, -\ e a i k o . Form of an x5 ^5 45 35 2? 35 #5 ^5 improved series; making seven in all, for the octave is but the same note with a different pitch. But upon what principle shall the values of these new sounds be determined, seeing we cannot have any Principles by ° _ which the series more simple consonances with the fundamental sound ^ improycd. whose vibrations are represented by 1 ? The answer is, we must take those sounds which make the simplest con- sonances while they give with the remaining sounds the greatest number of consonances. The consonance indicated by the interval from the 4th tO 8th is 2 - f = | ; Consonances; 5th to 8th is 2 ^ | = |; 4th to 6th isf -r f = f, Now, let the sound x, have between it and the 5th an First interval equal to that between the 5th and the 8th; then suPP^tion; wrill 3 _^ ™ — 4 a" • ^ 3^ whence x = 9 . Consequence; 8" 5 First three and the first three sounds of the series will stand 1, f, f, SSui; 142 NATURAL PHILOSOPHY. Intervals. Second supposition; giving intervals f and \°, already found in another part of the series. Again, let the interval between the 5th and 7th be equal to that between the 4th and 6th, and we shall have y • 2 45 and Consequence; y — 15 . 8 5 Lastthreesoundsttllsmakingthelasttliree sounds h V and 2, and giv- and their ing the consecutive intervals of f and }£, both of which are found in another part of the series. Replacing x and y by their values, we have Diatonic scale; Its effect universally agreeable; Note, its name, and place indicated; Illustrations; C,D,E,F,G,A,B,C 1 ^ ± 5 8 1 i_ J* L 5 4 5 3 5 2 5 3 1 5 5 8 , 2, or multiplying by 24 24,27 ,30 , 32 ,36 ,40 , 45 ,48, 1st, 2d , 3d , 4th, 5th, 6th, 7th, 8th. This is called the natural, or diatonic scale. When all its sounds are made to follow each other in order, either upwards or downwards, the effect is universally acknowledged to be pleasing, and all civilized nations have agreed in adopting it as the foundation of their music. Each sound in the scale is called a note, and takes the name of the letter immediately above it; and its place in the order of acuteness from the fundamental note is expressed by the ordinal number below it. Thus, count- ing the vibrations of the fundamental note unity, the note whose vibrations are f, is named E, and it is a third above C, regarded as the fundamental note; in like man- ner the note whose vibrations are f, is A, and it is a sixth above C. The 8th from the fundamental note, or C, is called an octave above C. Again, we say A, is a fourth above E, and E, is a fourth below A, as would be ELEMENTS OF ACOUSTICS. 143 manifest by simply sliding the scale to the right or in- Confirmedb verting it, so as to bring the number 1, under the note reference to the of reference regarded as the fundamental note. scalQ' § 139. This diatonic scale, which is obtained from the Essential series of sounds affording the simplest concords with the Q»alities of thQ fundamental note, after one alteration on account of the too great proximity of two concordant notes, and one in- terpolation on account of the too great distance of two others, has both of the essential qualities of repetition and variety. Thus, writing CD, for the interval from C to D, and using like notation for the others, and writ- ing the names which have been adopted by musicians for the several intervals, we have the following TABLE. C D = F G = A B . . = f; major tone. DE=GA .... — V;minortone, ffofamajor. E F= B C' . . . . ~\\\ diatonic semitone. CE=FA = GB . = f ; major third. E G— A Cr . . . . = |; minor third. DF......=ff; ff of minor third. CF=D G = EA =GC'=±; fourth. Table of intervals F B - 4 5 • flattened fifth * f°rmed fr°m the r ^.......—325 narxeneu nun. diatonic scale. C G = EB=FC . . =#: fifth. 2 5 DA ......= 40. so 0f a fifth. C A— D B . . . . = f; sixth. EC'......= |; minor sixth. C B.......= V 5 seventh.* DC......= '/ ; flattened seventh.f G C'.......= 2 ; octave. We observe here three different intervals between con- secutive notes, viz.: first, that from C to D = F to G * An inharmonious interval when the notes are sounded together. t Decidedly more harmonious than the seventh. 144 NATURAL PHILOSOPHY. Major tone; Minor tone ; Diatonic semitone or limma Least number of vibrations to produce continuous sound; Greatest number; Peculiarities of certain individuals ; Causes which render sounds inaudible. Diatonic scale may be continued in both directions; Practical limits. = A to B = |, and called a major tone ; second, that from D toE~ G to A = V°5 called a minor tone; and third, that from E to F = B to C = }J, called, though improperly, a diatonic semitone, being in fact much greater than half of either a major or minor tone. This interval is also called by some authors a limma. § 140. When the vibrations are less numerous than 16 a second (M. Savart says 7 or 8), the ear loses the impres- sion of continued sound, and in proportion as the vibra- tions increase in number beyond this, it first perceives a fluttering noise, then a quick rattle, then a succession of distinct sounds capable of being counted. On the other hand, when the frequency^of the vibrations exceeds the limit of 24000 a second, all sensation, according to M. Savart, is lost; a shrill squeak or chirp is only heard, and according to the observation of Dr. Wollaston, some in- dividuals, otherwise no way inclined to deafness, are alto- gether insensible to very acute sounds, while others are painfully affected by them. It is probable, however, that it is not alone the frequency of the vibrations which ren- ders shrill sounds inaudible, but also the diminution of intensity which, from the nature of sounding bodies, must ever accompany a rapid vibration among their elements. No doubt if a hundred thousand hard blows per second could be regularly struck by a hammer u x>n an anvil at precisely equal intervals, they would be heard as a deaf- ening shriek; but in natural sounds the impulses lose in intensity more than they gain in number, and thus the sound grows more and more feeble till it ceases to be heard. If we add to the diatonic scale on both sides the octaves of all its tones, above and below, and again the octaves of these, and so on, we may continue it inde- finitely upwards and downwards. But the considera- tions above show that we shall soon reach practical limits in both directions, growing out of the limited powers of the ear. ELEMENTS OF ACOUSTICS. 145 §141. Bv the aid of the ascending and descending Pioces of mil8ic ,. " . _ _ . & & played; series ot sounds thus obtained, pieces of music which are perfectly pleasing may be played, and they are said to be in the key of that note which is taken as the Key; fundamental, sometimes called the tonic note* of the Tonic note; scale, and of which the vibrations are represented by 1. And if such pieces be analyzed they will be found to consist chiefly if not entirely of triple or quadruple combinations of several simultaneous sounds called chords, chords; such as the following:: C,D,E,F,G,A,B ,C ,D',E-',F',G\A',B',C" 1 1 5 4 3. 5. 1_5 Q 18 10 8 6 1 0 3 0 J_ "^5 85 45 3525 3? 8 5^ 5 8 5 4 5 3" 5 "2 5 IT 5 8" 5 ^ 1st, 2d , 3d , 4th, 5th, 6th , 7th , 8th , 9th , 10th, 11th ,12th , 13th, 14th , 15th. 1st. The common or fundamental chord, called also chord of the the chord of the tonic, which consists of the 1st, 3d andtonic; 5th; or the 3d, 5th and octave. This is the most harmo- nious and satisfactory chord in music, and when sounded the ear is satisfied and requires nothing further. It is, therefore, more frequently heard than any other, and its continued recurrence in a piece of music determines the key in whicli the piece is played. 2d. The chord of the dominant. The fifth of the key chord of the note is called the dominant, by reason of its oft recur-dominant; ring importance in harmonic combinations of a given key. The chord of the dominant is constructed like that of the tonic, but on the dominant as a fundamental note, and consists of the 5th, 7th and 9th, being the 5th and 7th of one scale, and the 2d on the next following scale of octaves; or, replacing the latter note by its octave be- low, the notes of this chord will be 2d, 5th and 7th. 3d. The chord of'the sub-dominant; that is, the chord Chordof the constructed upon the 4th note next below the dominant. It consists of the 4th, 6th and 8th; or, replacing the lat- ter note by its octave below, the notes of this chord be- come the 1st, 4th and 6th. 10 146 False close. NATURAL PHILOSOPHY.' Dissonance of the 7th. 4th. The false close, which is the chord of the 6th, its notes being 6th, 8th and 10th, or replacing the last two notes by their octaves below, 1st, 3d and 6th. The term false close arises from this, viz.: A piece of mu- sic frequently before its termination (which is always on the fundamental chord) comes to a momentary close on this chord, whicli pleases only for a short time, and requires the strain to be taken up again and closed as usual, to give full satisfaction. 5th. The dissonance of the 7th, or the combination of the 2d, 4th, 5th and 7th. It consists of four notes, and is the common chord of the dominant with the noto immediately below it, or the 7th in order above it. short pieces of § 142. With these chords and a few others, music may be arranged in short pieces so as to possess considera- ble variety, but long pieces would appear monotonous. In the latter the fundamental note would occur so often as to appear to pervade the whole composition, and the change of key ear would require a change of key to avoid the feel- or modulation, j 0f tedium which would naturally arise from such a avoids monotony; ° ^ J cause. This change of hey is called modulation. But the change is not possible without introducing other notes than those already enumerated. Suppose, for example, it were desirable to change from the key of C to that of G. The chord of the tonic in the key of (7 is composed of the notes CEG; in the key of G, of the notes G B Df, giving the intervals, Example for illustration; CE=GB=i and CG=GD'=l In the chord of the dominant, in the key of C, the notes are GBDf, giving the intervals, GB=\, and GD' = %, the same as before. But the chord of the dominant in the key of G, if it could be formed at all from existing ELEMENTS OF ACOUSTICS. 147 notes, would consist of D',F',A', giving the intervals Necessity for other notes, shown ; &' F' = f f, and Df A' = \f, which are very different from the intervals of the com- mon chord to which they ought to be equal; and in • order that we may be able to make them equal, we must have other notes for the purpose. Now J?', being the dominant of G, must be the com-what the new mencement of the interval, and cannot be altered: not,es mUftt ' 7 replace; new notes must, therefore, be substituted for Fr and Af. Denote the vibrations of the new notes by x and y ; then, passing to to the octave below to avoid the com- mon factor 2, we must have, X 1 n-nrl V 3. To find the new — _ 4,«um _ __ ^ 5 notes; whence, substituting the value f for D, x — f . f- and y = f . f. That is to say, a change from the key of G to that of G, requires for the formation of the chord of the dominant in the latter key, two new notes, whose vibra- tions would be represented respectively by the ratios f- and | multiplied by the vibrations in the dominant of C. Now, as any note may be taken as the key note, and Multi licit f as the dominant changes with the latter, the number new notes must of requisite notes wTould be so numerous as to render the generality of musical instruments excessively com- plicated and unmanageable. It becomes necessary, there- fore, to inquire how the number may be reduced, and what are the fewest notes that will answer. For this purpose we remark, that if we multiply the values of x and y by 24, to reduce them to the same unit as that of the scale of whole numbers in § 18S, we find 148 NATURAL PHILOSOPHY. Uow this is accomplished; 24 sb = 9 8 ' 5 4 ' 24 = °64 1 24 y = 9 8 3 ' 2 •24 = 40i. Place of first new note determined; Sharp, flat; Place of second new note; In the scale just referred to we find the numbers 32 and 36, so that the note whose vibrations are x, is almost half way between these two notes, and may be in- terpolated at that place. It will, therefore, stand between i^and G, and is designated in music either by the sign #. sharp, or b, flat, according as it takes the name of the first or second of these letters. Thus it is written either *F, or b#. With regard to the note whose vibrations are y, and of which the value is 40{, it comes so near to the note A, whose value in the same scale is 40, that the ear can hardly distinguish the difference between them, 60 that the latter may be used for it; and though a small error of one vibration in 80 is introduced in using A as the dominant of D, yet it is not fatal to harmony, and it is far better to encounter it than to multiply pipes or strings to our instruments for its sake. Besides, these errors are modified and in a great measure subdued, by Temperament what is called temperament, of whicli the foregoing is the origin. § 143. The highest note of the perfect chord of the dominant of G, is three perfect fifths above C, and the note A', which we have adopted in its place, is the oc- tave of the 6th above C. The vibrations of the first are denoted, by (f)3 = y, and of the second, by 2 . f = V° 5 and the interval between will be 2_7 _1_ 1_0 — 8 1 8 * 3 8 0* commain music This interval of two notes, one of which rises three per- fect fifths, and the other an octave of the 6th above the same origin is called, in music, a comma. ELEMENTS OF ACOUSTICS. 149 § 144. Were any other note selected for the fundamen-No two keys -. • *i 1/ ill i j give the same tal one, similar changes wxmld be required; and no twoscaIe. keys can agree in giving identically the same scale. All, however, may be satisfied by the interpolation of a new note within each of the intervals of the major and minor tones in the scale of article (138), thus, *C) *D) #Fj #G) *A) Interpolation C, Or > , D, Or V *E, Fy Or } , G, Or V A , Or V B, G ; required; \>D) y>E)' *G) \>A) t>B) and the scale thus obtained is called the Chromatic chromatic scale; scale. § 145. But what shall be the numerical values of the Numerical values interpolated notes ? If it were desirable to make the of.the, * A ■*■ interpolated scale of article (138), which is in the key of C, (the vi- notes; brations of this note being represented by unity,) as per- fect as possible, at the expense of the others, there would be but little difficulty, as the mere bisection of the lar- ger intervals would possibly answer every practical pur- pose, and #C~ ^D, might be represented by VI -f; #D = t>E, by V f • f-, and so on ; but as in practice no such preference is given to this particular key, and as variety is purposely studied, we are obliged to depart from the pure and perfect diatonic scale ; and to do so Necessity for & with the least possible offence to the ear, is the object of tlm^ament a system of temperament. If the ear required perfect concords, there could be no music but a very limited and monotonous one. But this is not the case ; per- fect harmony is never heard, and if it were, would be appreciated only by the most refined ears; and it is this fortunate circumstance which renders musical com- Perfect harmony position, in the exquisite and complicated state in which never hcard' it at present exists, possible. § 146. To ascertain to what extent the ear will bear a departure from exact consonance, let us see what takes 150 NATURAL PHILOSOPHY. Explanation; Extent to which place when two notes nearly but not quite in unison Xrrtrefrom or concord are sounded together. Suppose two equal exact and similar strings to be equally drawn aside from their consonance; pOSitions of rest and abandoned at the same instant, and suppose one to make 100 vibrations while the other makes 101, and that both are at the same distance from the ear. Their first vibrations will conspire in produc- ing sound waves of double the force of either singly, and the impression on the ear will be double. But on the 50th vibration, one will have gained half a vibra- tions, and the motions of the aerial molecules produced by the co-existing waves from both strings will no lon- ger be in the same but in opposite directions; and this being sensibly the case for several vibrations, there will be an interference and a moment of silence. As the vibrations continue, there will be a further gain, and at the 100th this gain will amount to one whole vibra- tion, when the waves will again conspire, and the sound have recovered its maximum intensity. These alternate reinforcements and subsidences of sound are called beats. Let n, denote the number of vibrations in which one string gains or loses one vibration on the other, m the number of vibrations per second made by the quicker, and t, the interval between the beats, then will Beats. Example for illustration. m : n :: 1* : t whence n from which it is obvious that the nearer two notes ap- proach to exact unison, the longer will be the interval between the beats. Effect of erfect §1^7. -^nd here it may be proper to remark upon concords; the effect produced in perfect concords and in those ELEMENTS OF ACOUSTICS. 151 only which are perfect. If one note make m vibrations These effects for " , _ . . ., - .„ , ., two concordant while another makes n, it is obvious that it the vibra-notesexplained tions begin together, the mth vibration of the one will and illustrated: conspire with the nth of the other, and the effect upon the car of these conspiring vibrations will be similar to that of a third set of which each individual vibra- tion conspires with every mth vibration of the one and every nth vibration of the other of the concordant notes. This third set will give rise to a note graver than either of the others, and its pitch will, Equation (41), be the same as that of a fundamental note of which the con- cordant notes may be regarded as harmonics. ThisEesultant aml J o components. graver note is called the resultant, and those from which it arises, components. Let m — Z and n = 2, then, see scale of article (138), will the concord be a perfect fifth, and the resultant note will be an octave below the gra- ver of the two components. What is true of two notes in perfect accordance may be Above shown to be equally true of several, and hence the ex- ~^r^verai planation of this curious fact, viz.: that if several strings concordant note- or pipes be so tuned as to be exactly harmonics of one of them, that is, if their vibrations be in the ratios 1, 2, 3, 4, &c, then, if all or any number of them be sounded together, there will be heard but one note, and that the fundamental note. For, all being harmonics of the note 1, if we combine them two and two we shall find compara- tively few but what will give resultants which, with the individual notes, will be lost in the united effect or re- sultant of all the component sounds. But to produce this effect the strings or pipes must be tuned perfectly to strict Explanation of harmonics. The effect can never take place on the strings of a piano-forte, since they are always tempered. § 148. Now, to resume the question of temperament: If we count the notes in the chromatic scale of article Temperament . .. resumed; (144), we shall find thirteen, and consequently twelve in- tervals. Hence, if we would have a scale exactly similar in all its parts, and which would admit of playing equally 152 NATURAL PHILOSOPHY. scale that would well in any key, the question of temperament would re- piay equally weii c|uce to ^a|. 0f inserting 11 geometrical means between in any key; the extremes 1 and 2, and the scale would stand, 11 1, 212,2T2,212, . . 212, 2. Ascending the scale by fifths; iso-iiarmonic The values of the mean terms are readily computed by scale. logarithms. This scale, which is one of perfectly equal intervals, is called the Iso-harmonic scale. Examination of g 14.9. If we examine the chromatic scale and table of ^ehaTteStoofintervals in article (139), we shall find that the interval intervals; from E to F, and from B to C, are semitones, and that in a perfect fifth there are, therefore, seven, and in an octave twelve semitones. If, then, we reckon upwards by fifths, we shall, after twelve steps, come to a note in the ascending scale of octaves of the same name as that from which we set out. Beginning with C, for example, we shall, after the twelfth remove, arrive at another C; or, which amounts to the same thing, if we ascend by two-fifths from C and descend an octave, we fall upon D; in like manner, rising by two-fifths from D and falling an octave, we fall upon E, and this pro- cess being sufficiently repeated, we finally reach G', the octave of G. xigain, from the same scale and table, we see that in a major third, that is, from C to E, there are four semitones, and hence, if we ascend the scale by major thirds we shall, after three steps, arrive at the octave of the note from which we started. The value of a fifth is f, of a major third f, and of an octave 2. Now, there is no power of § or of £ equal to any power of 2, and hence there is no series of steps by perfect fifths or major thirds that can lead to any one of the octaves of the fundamental note. Were the chromatic scale perfect, twelve perfect fifths should be equal to seven octaves, and three major thirds to Ascending by major thirds. Value of a fifth, of a major third, and of an octave. ELEMENTS OF ACOUSTICS. 153 one octave; but, as just remarked, neither of these can Reckoning be true of perfect fifths or major thirds, for (§)12 =129,74, n^thirds, or and 27 = 128, giving a difference of nearly one vibra- by fifths; tion in every 64 ; and (f)3 = 1,953, instead of two. So that, if we reckon upwards by major thirds, we fall con- tinually short; if by fifths, we surpass the octave. The excess in this latter case is called the wolf, a name sug-The wolf; gested, no doubt, by the fact that the thing whicli bears it has been hunted and chased througli every part of the scale in the vain hope of getting rid of it. In con- sequence, it has been proposed to diminish all the fifths equally, making a fifth, instead of f, to be equal to 2T2, and tuning regularly upwards by such fifths, and from the notes so tuned, downwards by perfect octaves. This frstem of e(Jual 1 u x temperament; constitutes what is called the system of equal temperament. In this system the notes must all be represented by identical with the _i_ .,,»..-. iso-harmonic; the different powers of 212, and the system itself is iden- tical with the Iso-harmonic. Theoretically, it is the sim- plest possible. It has, however, one radical fault; it o-ives all the keys one and the same character. In any its defects, and other system of temperament some intervals, though of c y' the same denomination, must differ by a minute quantity from each other, and this difference falling in one part of the scale on one key and in a different part on an- other, gives a peculiar quality to each, and becomes a source of pleasing variety. Some have supposed that temperament only ajrplies to General instruments with keys and fixed notes. This is a «n^-^0^"^; take. Singers, violin players, and all others who can pass through every gradation of tone, must all temper, or they could never keep in tune with each other, or with themselves. Any one who should keep ascending by fifths and descending by thirds or octaves, would soon find his fundamental pitch grow sharper and sharper, till he could neither sing nor play; and two violin players accompanying each other and arriving at the illustrations. the same note by different intervals, would find a con- tinued want of agreement. V 154 NATURAL PHILOSOPHY. Construction of a table; § 150. If we take the logarithms of the fractions which express the intervals from the fundamental note to that of any other in the diatonic scale, we shall find, after multiplying each logarithm by 100, to avoid fractions, taking the product to the nearest whole number, and then the successive differences between these, the following TABLE. Table; From Cto C Cto D Cto E CtoF Cto G Cto A Cto B Cto C Intervals. Ra- tios. 0 major tone major third minor fourth major fifth major sixth major seventh octave 9 8 5 4 1 3 3 2 5. 3 L.5 8 2 Logarithms- T-- 0,00000 0 0,05115 51; 0,09691 0,12494 0,17609 0,22185^22 0,27300 0,30103 Ap- pro*. Differences. 125 176 273 301 2S=0 = 51 = T= 46=* = si=r= 28=0 = maj. tone minor tone limma maj. tone ■minor tone maj. tone limma Its accuracy; Its use; Three intervals and their notation: Enharmonic diesis; The approximate values for the intervals are true to the 500th of a tone, an interval far too small for the nicest ear to distinguish; these values may, therefore, be used in all musical calculations when no very high powers of them are taken. Since the logarithm of any interval is equal to the logarithm of the higher, diminished by that of the lower note, the numbers in the column of differ- ences may be taken to represent the values of the se- quence intervals, or intervals between the consecutive notes expressed in equal parts of a scale of which it takes 301 parts to measure an octave. And we here perceive again the three different kinds of intervals referred to in article (139). They are de- noted in the table above by the characters T, t, and 0, their values being respectively 51, 46 and 28, corres- ponding to the fractions f, \° and if of the article just cited. These intervals give rise to what is called the enharmonic diesis, whicli is the interval between the sharp of one note and the flat of that next above it, and enables us to understand the distinction between ELEMENTS OF ACOUSTICS. 155 flats and sharps; a distinction essential to perfect har- mony, but which can only be maintained in practice in organs and other complicated instruments which ad-its use; mit of great variety of keys and pedals, or in stringed instruments or in the voice, where all gradations of tone may be produced. To understand this distinction, suppose in the courseTo modulate of a piece of music it be desirable to modulate from another. the key of C to that of F, its subdominant. To make the new scale of F perfect, its intervals should be the same and succeed each other in the same order as in the original key of C. That is, setting out from F, the sequence of intervals should be T t 0 T t T 0, as in the table. Kow, this sequence does not take place in the unaltered scale of C, when we set out from any note but C, and if we prolong this scale backward to F, the notes will stand F G A B G D E F' T t T 6 T t 6 ~y _^v_ "Y~ Notes as they stand erroneously; whereas they should stand, F G A B C D E F! T t 6 T t T 6 y\. ~Y~ y^ ~r~ Notes as they should stand; The first two intervals are the same in both. The next First two .-,, .P n ., ,i i.T> j. 4. intervals same in two will agree if wre flatten the note B, so as to invert both. the intervals, or make, \>B - A = 6 - 28; To make the next two agree; and G- bB = T= 51; 156 NATURAL PHILOSOPHY. Supposition; giving by addition G- A = T+0= 51+28 =79= major third. The quantity by which B must be flattened for this purpose is obviously Consequence; T - 6 = 51 - 28 = 23; Interpolation necessary to render the two scales nearly perfect in one particular case. Another case supposed; and this is the amount by which, in this case, a note differs from its flat. As to the remaining three inter- vals, the difference between Tand t being small, amount- ing only to 5, (which answers to the logarithm of a comma ££,) the sequence Tt 0 is hardly distinguishable from t TO, and if the note D be tempered flat by an T—t interval = —^-> or half a comma, this sequence will in both cases be the same, and our two scales of G and F will be rendered as perfect as the nature of the case will permit by the interpolation of only one new note. But, on the other hand, suppose we would modulate from G to B. In this case the scale of C will stand Scale as it stands Jlj (j in this case; D E F G T A B' T whereas it should be Scale as it should fctand; B *C *D E #F #G #A B' T t 6 T t T 6 Conclusions. The intervals from B to E, and from E to B, are the only ones that are equal, and to make the others equal would require C, D, F, G and A to be sharpened, and consequently the introduction of no less than five new notes. to #A we have But to confine ourselves to the change from A ELEMENTS OF ACOUSTICS. 157 JS — A = T = 51 " Particular case taken. and B - *A = A = 28; consequently, by subtraction, A - *A = 23 = B - *>B, Result; as before determined. But since the whole interval from B to A = T = 51, is more than double this interval, the flattened note bj5, will lie nearer to B, and the Explanation; sharpened note %A nearer to the lower one A than a note arbitrarily interpolated half way between A and B, (to answer both purposes approximately,) would be, Diesis left in and thus a gap or diesis, as it is called, would be leftthis case; between #A and ^B. The diesis in this case onlv amounts to T-2 (T- d) mat u amounl3 v J to. = 51 — 46 = 5, equal to a comma, or the tenth part of a major tone I1; in other cases it would be greater. But in all cases the interval between any note and its sharp is considered to be equal to that between the , same note and its flat. §151. Taking each note of the diatonic scale as the Each note of the /» -i it • • i n n -i diatonic scale fundamental or key note in succession, we shall nnd, takenastnek by the same mode of comparison, the following sets ofn°te; notes in the several scales—the accent at the top of the letter denoting one octave above the key note. Names of tho Keys. C, I>, E, F, G, A, 2?, C. (natural, C) B, E, *F, G, A, B, *C, D'. (two sharps, D) setsornot« E, »F, *G, A, B, *C, #B, E'. (four sharps, E) thns*"u,d' F, G, A, i>B, C, B, E, F'. (one flat, F) G, A, B, C, Z>, E, #F, G'. (one sharp, G) A, B,*C, D, E,*F,*G, A', (three sharps, A) B, *C, #D, E, *F, *G, *A, B'. (five sharps, B) 158 NATURAL PHILOSOPHY. These scales defective by two sharps; Result of a particular supposition. In these scales which have the natural notes of the diatonic scale for the key, there are but five sharps, whereas there should be seven. Where are the other two? If we take *F and #67 as the key notes, we shall find Names of the Keys. %F,*G, *A, B, *C, *D, %E, #F'.(six sharps, #F) *C, *D, *E, *F, *G, *A, #B, *Cf. (seven sharps, *C ) In like manner, constructing a diatonic scale on t>i?, and on each new flat as it is successively introduced, we find the following, in which the accent at the bottom of a letter denotes one octave below the kev. Same for another b JfP supposition. , . \>A, bD t>G *>C Karnes of the Keys. C, B,i>E, F, G, A, bB. (two flats, \>B) F, G,bA,bB, C, D\ i>E'. (three flats, bE) bB„ C ,bZ>,t>F, F, G,bA. (four flats, bA) bE, F, bG, M, bB, C, bD'. (five flats, bD) M, bB, bC, bB', bF\ F\ bG'. (six flats, bQ) l>2>, *E, *>F,bG,bA,bB, bC. (seven flats, bQ) th'vised; several systems §152. Assuming the principle that the interval of temperament -, , have been between any note and its sharp is to be equal to that between the same note and its flat, a variety of systems of temperament have been devised for producing the best harmony by a system of twenty-one fixed notes, viz: the seven notes of the diatonic scale with their seven sharps and seven flats. Among the most remarkable sys- sy(';trsmarkable temS may be mentioned those of Huygens, Smith, Young and Lagier, for an account of which the reader is referred to the Encyclopoedia Metropolitan, article, Sound. Vol. IV., page 797. Some of the Peculiarity of the piano-forte. 153 But the piano-forte, an instrument in almost universal use, and of the highest interest to all lovers of music, admits of only twelve keys from any one note to its octave, and a temperament must be devised which will accommodate itself to this condition. ELEMENTS OF ACOUSTICS. 159 We have already spoken of the division of the octave Arguments in . favor of equal into twelve equal parts, and have seen that this makes tenipcrament. the fifths all too flat, the thirds all too sharp, and gives a harmony equally imperfect in all the keys. It is urged in favor of equal temperament that all the keys are made equally good, and that in no one does the temperament amount to a striking defect; also, that in the orchestra there is little chance of any uniform temperament if it be not this. Against equal temperament it is urged, how- ever, as before stated, that it takes away all distinctive Against equal character from the different keys, and after all, leaves no tcmPerament one of them perfect. A piano-forte perfectly tuned by the system of equal temperament has to some persons a certain insipidity which only wears off as the effect of this tuning disappears ; insomuch that the best phase of illustration by •^ o J. I 7 -i ^ the piano-forto the instrument is exhibited during the period which pre- cedes its becoming disagreeably out of tune, or, more properly, while it is assuming a state of maltonation ; for, the transition is only a change from equal to unequal temperament, in which the several keys begin to exhibit variety of character, until maltonation arrives and makes the instrument offensive. The best practicable way of obtaining a given tempera- use of the ment, equal or unequal, is by means of the monochord. raonochord; The proper lengths of the strings of this instrument, to form the required notes, are first calculated, and after- wards those of the instrument to be tuned are brought into unison with them. No tuner can get an equal tempera- ment by trial; so that the question in practice generally General aim in lies between all sorts of approximations to equal tempera- practice. ment, and as many approximations to some other tem- perament. 8 151. The mode of proceeding by approximation to The most usual equal temperament is simply to tune all the fifths a little proceeding; flat; and the following order is the most usual. The first letters represent the note already tuned, the second the one which is to be tuned from it; a chord in parenthesis 160 NATURAL PHILOSOPHY. First step, by represents a trial that should be made on notes already tuning fork; tuned, to test the success of the operations as far as it has gone. The first step is to put C' in tune by the tuning fork ; «*««-, <*'; C'C; CG; GO,; G,D; DA; AA,; A,E; (CFG); EB; {CFG; DOB); BB,; B,*F; (D#FA); *F#F/, #F,*C: {A,*C E); #C*G; (E#GB); C'F; (FAC); F*At; (*A,DF); #At #A ; (*At *D; (#27 G *A); *D*G,; {*G, C #27). Explanation of method, and of results that should be obtained ; Bearinsrs. All the semitones are written as sharps whether tuned from above or below. Since the fifths are all to be a little too small in their intervals, the upper notes must be flattened when tuned from below, and the lower notes sharpened when tuned from above. In the preced- ing, the octave C C is completely tuned, and also the adjacent interval #Ft C. The rest of the instrument is tuned by octaves. The thirds should come out a little sharper than perfect, as the several trials are made, and when this does not happen, some of the preceding fifths are not equal. The parts which are first tuned by fifths, and from which all the others are tuned by octaves, are called bearings. Remark on § 155. In unequal temperament, some of the keys are *q,u . kept more free from error than others, both for the temperament; x " sake of variety and because keys with five or six sharps or flats are comparatively but little used; these latter keys are left less perfect, and this is called throwing the toolf into these keys. From equal intervals to those which produce what has been called maltonation, there is abundant room for the advocates of unequal tempera- ment to select that particular system most congenial to the views of each, and, accordingly, many systems have been proposed. Of these we shall only mention two, smith's system: viz.: that denominated by Dr. Smith the system of mean ELEMENTS OF ACOUSTICS. 161" tones, and that which bears the name of its author, Dr. Young. The system of mean tones supposes the octave divid- s^stem of mean ed into five equal tones, of which we shall denote the ' value of each by a, and two equal limmas, each hav- ing the value j3, succeeding each other in the order a apaaafi instead of Tt d Tt T&, as in the diatonic scale, and such that the thirds shall be perfect, and the fifths tempered a little flat. These conditions are sufficient to determine the values of a and (3, for, 5 a + 2 0 = 1 octave = 3 T + 2t + 2& 2 a = 1 third = T + t whence T+t T-t a 2 and substituting the values from the table Use of this • Q =— Q J_ ------- *, system explained and illustrated; a 51 +46 51 - 46 2 = 48,5 ; 0 = 28 +---j---= 28,125 and since the interval from the 1st to the 5th of the scale is 3a + P = 2T+t + & - 4 ' the fifth by this scale is flatter than the perfect fifth by the quantity j (T— t), that is, by a quarter of a com- Results. ma. In this system the sharps and flats are inserted by bisecting the larger intervals. Dr. Young's first system is as follows, viz.: Tune Youn^s first downwards from the key note six perfect fifths, then up-system; wards from the key note six imperfect fifths, dividing the excess of twelve perfect fifths, above seven octaves, li 162 NATURAL PHILOSOPHY. Explanation. Scale of tho Chinese, Hindoos, &c. Effect of small intervals. equally among the imperfect fifths, and observing to as- cend in the first case, and descend in the second, by octaves, when necessary, to keep between the key note and its octave. § 156. If we take from the diatonic scale the notes F, and B, which rise from those immediately preceding them by semitones, there will remain C, D, E, G, A and C for all the sounds of the octave. This is the original scale of the Chinese, Hindoos, the Eastern Islands and the nations of Northern Europe. It is the scale of the Scotch and Irish music, and the Chinese have preserved it to the present time. The character of this scale is exhibited by playing on the black keys alone of the piano-forte. § 157. The effect of making an interval smaller is to give the consonance a more plaintive character. It may easily be observed, for example, that the intervals of the minor third, E G, and minor sixth, E C' on any instru- ment, have a sad or plaintive effect as compared with the major third, GE, and major sixth, C A. Almost all per- music applied in sorig jn ordinary conversation are constantly varying the tone in which they speak, and making intervals which approach to musical correctness, and the effect of sorrow, res-ret, and the like, is to make these intervals minor. Anv one with a musical ear, noticing the method of say- ing " / cannot? pronounced as a determination of the will, and comparing the same uttered as an expression of regret for want of ability, will understand what is here meant. Why this is so, no one can tell. But the asso- ciation exists, and resort is had to those modifications of the diatonic scale which are known from experience to produce the emotions here referred to. The results of these modifications, of wdiich there are several, are called Minor Scales, in contradistinction to the diatonic, which is called the Major Scale. The change from a minor to the major scale is one of the most effective of musical resources. Principles of Minor scales. ELEMENTS OF ACOUSTICS. 163 If we return to the fundamental note C and its conso- nances, viz.: C t>E E F G A G' Fundamental 1 6 5 4 3 5 Q. noteandits x 5 55453525 35-'5 consonances; and instead of rejecting ^>E as too near to E, we discard this latter note, and finish by inserting D and B of the diatonic scale, we shall have what is called the common ascending minor &cale, as follows: C , D ,^>E , F , G , A , B , C Ascending minor 11 ? 7 7 ' 7 scale; 1 9 6 4_ 3. 5. 1_5 Q -L5 8*5 55 352535 85^* But it is not easy to recognize this as a minor scale in Not easily descent, because, in going from Cf to C, there is no dis- ^^clie in tinction between it and the major scale till we come to Ascent; bE, or until the scale has produced its principal effect upon the ear. To remedy this, A and B are both lowered a semitone ; that is, A is made bA, and B is made *>B, thus making *>A a fourth to *>E, and ^>B a fifth to *>E, and giving C,D,t>E,F,G,bA, t>B,C' 1 9 642 1 9_9v* ±58"5 5 5 3" 5 2 5 55 55^5 which being reversed, is called the common mode of Descending the -, 7 • i i • i minor scale. descending the minor scale. Again, if we retain B of the major scale and lower A, we have C,D,bJS,F,G,*A,JB9C' 1 9 6 4 3 1 1_5 Q J-5"8"5 5 5 3" 5 2 5 558 5 ^5 which is a mild and pleasing scale both in ascent and Schneider,s descent, notwithstanding the wide interval between b A principal minor and B. Its harmonics are more easy and natural thanscale' the other, and Schxeider makes it, in his Elements of 161 NATURAL PHILOSOPHY. Harmony, a principal minor scale, and treats all others as incidental deviations. Any system of §158. We shall now show how we may, from the mlyTe^mined theory of the scale, examine any system of tempera- by the scale; ment; and as the method will be rendered the more obvious by applying it to a particular example, we shall take the system of Dr. Young just described. Let all the intervals be expressed in mean semitones, as the unit. There being twelve semitones in the oc- tave, we have one semitone equal to the logarithm of 2 divided by 12, or °>030103 = 0,0250858: 12 Method explained; System of Dr. Young taken; Example for illustration; and dividing the logarithm of the major tone = \, that of the minor tone = V°5 that 0I> the diatonic semitone = if, and the excess of twelve perfect fifths over seven octaves = 0,00588 by this value of the mean semitone, we shall find 1 major tone = 2,039100 mean semitones, 1 minor tone = 1,824037 1 diatonic semitone = 1,117313 Excess of 12 fifths over 7 octaves = 0,231600 u (C it It u In tuning upwards, each fifth is to be flattened by one-sixth of 0,234600, or by 0,039100. In the equal tem- perament the wolf is replaced by twelve equal whelps; here by six, but of double the size. Now, a perfect fifth is composed of 2 major tones 1 minor tone 1 diatonic semitone Perfect fifth Deduct . . = 4,078200 = 1,824037 = 1,117313 = 7,019550 0,039100 Flattened fifth = 6,980450 ELEMENTS OF ACOUSTICS. 165 Then taking C for the key note, C -5th . . 12,00000 . . 7,01955 C . +5(A . . 0,00000 . 6,98045 Example continued; F . . 4,98045 •(1) 0 . . 6,98045 . (i) +8th . 12 +5 . D' . . 6,98045 F' . . 16,98045 . 13,96090 -5'* . . 7,01955 . . 9,96090 .(2) -8'* . D . . 12 #A . 1,96090 . ■(2) -5 . 8,94135 . (3) +8'* . 12 +5'* . E' . . 6,98045 #2?' . . 14,94135 . 15,92180 -5'* . .-7,01955 . (5) -8th . E . +5(A . B . . 12 #G —5th . . 7,92180 . . 7,01855 . . 0,90225 . 3,92180 . . 6,98045 • (4) ^ ' The same; #C . 10,90225 . . (5) +8'* . 12 +5'* . *F' . . 6,98045 #c . . 12,90225 . 17,88270 -5th . . 7,01955 . (6) -8tk . #F . . 12 #F . . 5,88270 . 5,88270 . . (6) Collecting these intervals for all the notes from C to C, we have C *c D #D E F 0,00000 #F 0,90225 O 1,96090 *G 2,94135 A 3,92180 *A 4,98045 B 5,88270 . 6,98045 7,92180 . 8,94135 9,96090 10,90225 Results collected. As the most important chord is that of the tonic, we form our idea of the effect of each key, from the effect of the temperament upon this chord, judging of the character of the key by the amount and direction of 166 NATURAL PHILOSOPHY. Explanation; the temperament upon the third and fifth, whicli with the key make, as we have seen, the chord in question. Now, a major third is composed of 1 major tone = 2,03910 mean semitones, 1 minor tone = 1,82404 " " Value of a major third; Major third . . 3,86314 A minor third is composed of u a Value of a minor third; 1 major tone diatonic sei Minor third . 1 diatonic semitone = 1,11731 " = 2,03910 mean semitones, 3,15641 " u Conclusions. Method of and hence the intervals for the chord of the tonic are For a major key . . 3,86314 and 7,01955 " minor " . . 3,15641 and 7,01955. To examine any particular key, take the numbers from the preceding table opposite the notes of the tonic chord, adding twelve to make the octave when necessary; sub- tract the number of the key note from each of the other two, and the remainders will give the tempered examining any m o i particular key. intervals ; from these remainders subtract the correct in- tervals above, and these second remainders will give the amount and direction of the temperament. For exam- ple, let us examine the key of A; we find A . 8,94135; #C . 12,90225; Ef . 15,92180 8,94135 8,94135 Tempered intervals 3,96090 . . . 6,98045 Perfect intervals . . 3,86314 . . . 7,01955 Temperaments . . + 0,09776 . . - 0,03910 whence we see that the first interval is sharper and the uS'oT SeCOnd flatter than Perfect> tlie sign +, indicating sharper, and the sign —, flatter. END OF ACOUSTICS. ELEMENTS OF OPTICS. § 1. The principle by whose agency we derive ourLight sensations of external objects through the sense of sight, is called light ; and that branch of Natural Philosophy which treats of the nature and properties of light, is called Optics. p 1CS § 2. There exists throughout space an extremely at- PriudPle of tenuated and highly elastic medium called ether. This ether permeates all bodies, and the pulsations or waves propagated through it, constitute the principle of light. The eye admitting the free passage of the etherealSensatlon of d ° j. o sight produced; waves into it, the sensation of sight arises from the motions which these waves communicate to cer- tain nerves which are spread over a portion of the internal surface of that organ; we therefore see by aAnal°sybetween ° */# the sensations of principle in every respect analogous to that by which Right and sound. we hear; the only difference being in the nature of the medium employed to impre'ss upon us the motions proper to excite these different kinds of sensations. In the former case it is the ether agitating the nerves of the eye, in the latter, the air communicating its vibra- tions to the nerves of the ear. § 3. Some bodies, as the sun, stars, &c, possess, in Self-luminous 0 7# 7 ... bodies; their ordinary condition, the power of exciting light, while many others do not. The first are called self- luminous, and the second non-luminous bodies. All substances, however, become self-luminous when their ^on;lummon8 " > bodies; temperature is sufficiently elevated, or when in a state 168 NATURAL PHILOSOPHY. Insects that possess the power of exciting light. Self-luminous bodies visible; N on-luminous rendered so. of chemical transition ; and some organisms, as the glow- worm, fire-fly, and the like, are provided with an appa- ratus capable of exciting ethereal undulations and of becoming self-luminous when thrown into a state of vibration by these insects. Self-luminous bodies are seen in consequence of the light proceeding directly from them; whereas, non-lu- minous bodies only become visible because of the light which they receive from bodies of the self-luminous class, and reflect from their surfaces. Medium. § 4. "Whatever affords a passage to light is called a medium. Glass, water, air, vacuum, &c, are media. Waves of light spherical in homogeneous media; Geometrical. illustration. Wave front mot spherical in heterogeneous media. Fig. 1. § 5. Waves of light, like those of sound, proceed from any disturbed molecule as a centre, with a constant velocity in all directions, through media of homogeneous density. The front of the luminous wave in such media is, there- fore, always on the surface of a sphere whose centre is at the place of primitive disturbance, and whose radius is equal to the velocity of propagation multiplied into the time since the wave began. Thus, if a molecule of ether be disturbed at C, and the velocity of propagation be denoted by V, and the time elapsed since the disturbance by t, then will the front of the wave at the expiration of this time be upon the surface of a sphere whose centre is at C and radius G A — V. t. If the medium through, which the wave moves be not homogeneous, the shape of the wave front will not be spherical, but will vary from that figure in proportion as the medium de- parts from perfect homogeneousness. § 6. The circumstances attending the propagation of luminous and sonorous waves are similar. The intensity ELEMENTS OF OPTICS. 169 media of light, like that of sound, depends upon, and is directly intensity of proportional to the amount of molecular displacement. Ife It is, therefore, Acoustics, § 53, inversely proportional to the square of the distance from the original luminous source. § 7. We have seen, Acoustics, § 16, that in wave pro-To demonstrate ,. .i ii the rectilineal pagation through a homogeneous me- .. , x o c? o propagation of dium, the displacement of a mole- Fig. 2. Msnt in -\ s\ r* ., i r* , 1 homcseneous eule (J, trom its place ot rest at one time, becomes a source of displace- ment at a subsequent time for an in- definite number of molecules situat- ed on the surface of a sphere M JV, whose centre is at 0, and of which the radius is equal to V. t; that these numerous disturbances become in their turn so many sources of disturb- ance for any single molecule as Of, in front of the wave, and that the amount of 0"s displacement from its place of rest will be found by compounding the displacements due to all these sources, after estimating the amount due to each separately. To ascertain the effect of this process of composition, Geometrical - . • • >o i construction and denote by \ the length ot a luminous wave; join (J and explanation ; Of by a right line, and take the distances A B = B C = CD — DE=\\ and with 0' as a centre and the distances 0' B, 0 G, 0' D, 0' E, &c, successively as radii, describe the arcs Bb, Gc, Dd, Ee, &c, cutting the section of the wave 31N, in the points b, c, d, e, &c. Now, regarding the several molecules in the por- tions Ab, be, cd,de, &c, of the great wave, as so many centres of disturbance, it is obvious that the secondary waves sent to the molecule Or, from those which occupy corresponding positions, on each pair of consecutive por- tions, will be in complete discordance, and therefore, Joint effects of Acoustics, § 59, that the joint effects of any two consecu- ^^17 tive portions will be to destroy one another, provided main wave; 170 NATURAL PHILOSOPHY. Portions of the main wave remote from the straight line destroy each other; Displacement of an assumed particle due to those portions of the main wave in the immediate vicinity of the right line joining it with the luminous origin; Portion producing the greatest effect; Effects of the other portions. Conclusion. V \ the waves from these portions be equal in number and give equal molecular displacements. And it is easy to see that this is the case with respect to the portions remote from A. For, the magnitude of the displacement of Or, caused by any two consecutive portions, depends— first, upon the relative magnitudes of these portions, and secondly, upon their difference of distance from 0'. With respect to the former, it is obvious, from the con- struction, that Ab is greater than be, be than cd, cd than de, and soon; but that the successive differences go on continually diminishing, and that the magnitudes of, and consequently the number of waves from, the succeed- ing portions, approach indefinitely to equality as they recede from the point A. For corresponding points in consecutive portions, the difference of distance, which is \ X, never exceeds, as we shall see, 0,000013 of an inch ; so that the portions of the main wave remote from the straight line O Gf, destroy each other's effects, and the displacement of O', will be entirely due to those parts of the great wave in the neighborhood of the line con- necting the point Of with the luminous origin. Of these parts A b produces, of course, the greatest effect, being both the largest and least oblique to O O'. The effects of the neighboring portions are, however, sensible, and we shall have occasion, under the head of chromatics, to observe some important phenomena to which they give rise. In the mean time we cannot fail to perceive one remarkable consequence of this explana- tion, viz.: that if the alternate portions be, d e, &c, whose effects are, relatively to the others, negative, be stopped, the total effect upon O' will be augmented, and the light there will be literally increased by intercept- ing a portion of the wave. All of which we shall have occasion to see fully confirmed by experiment. For the present our conclusion is, that in a homogeneous me- dium, the apparent effects of light are propagated from one point to another in a right line; that the sensible effects of light cannot, like those of sound, be propa- ELEMENTS OF OPTICS. 171 gated round corners, and that optic shadows must run ^t not up to the right line drawn from the luminous source ^^"^ tangent to the edges of objects which cast them. § 8. Any line JR R, which pierces the wave surface perpen- dicularly, is called a ray of light. A ray, therefore, is obviously a line along which the successive effects of light occur. When the wave surface be- comes a plane, the rays will be parallel, and a collection of such rays is called a beam of light. When the wave surface is spherical, the rays will have a common point at the centre of curvature, and a collection of such rays is called a pencil of light. Eay of light. JL Beam of light =:^j> Pencil of light. REFLEXION AND REFRACTION OF LIGHT. § 9. The reciprocal action between the molecules of Reflexion and various substances and those of the ether which pervades ^h*c lor them, causes the latter fluid to exist in a state of different elasticity and density in different bodies. By reference to Equation (3), Acoustics, we recall that the wave velocity increases with the elasticity of the medium and decreases with its density; and, § 71, same subject, shows us, that when a wave is incident upon the boundary of a medium of different density from that in which it is moving, it will be resolved into two component waves, one of whicli will be driven back from the bounding surface, while the Follows the sam« other will be transmitted and conducted through the ne\vlawsassound; medium. Light, like sound, will, therefore, be reflected and refracted, and according to the same laws. 172 NATURAL PHILOSOPHY. And the circumstances of incident and deviated light determined by the same equation. 10. And resuming Equation (29), Acoustics, which is V . sin.

S" by refraction ; also drawing the normal PPr to the deviating surface, the angle P D S, which the incident ray makes with this normal, is called the angle of incidence; the angle P D S', which the reflected ray makes with the normal, is called the angle of reflexion, and the angle PDS"' = P'DS", which the refracted ray makes with the normal, is called the angle of refraction. How these angles are estimated; § 12. These angles are always estimated from that part of the normal drawn through the point of incidence of the ray, which lies in the medium of the incident wave. ELEMENTS OF OPTICS. 173 Fig. 5. Illustration. They are accounted positive when on the same side of when positive the normal as the incident ray, and negative when on™jjjj**° the opposite side. Thus, the angle of incidence P D S, is always positive, as1 also the angle of re- fraction PDSm, while the angle of reflexion PDS', will always be negative, as it should be, since the velocity of the reflected light must be counted negative, the reflected wave being dri- ven back from the "de- viating surface. § 13. When the deviating surface is curved, we con- ^^ eviatlD8 ceive a tangent plane drawn to it at the point of incidence, and treat this plane as the deviating surface for that portion of the wave which is incident immediately about the tangential point. § 14. The angle which any ray after deviation, makes with the prolonga- tion of the same ray be- fore incidence, is called the deviation. Thus, Slv- D S,' is the devia- tion by reflexion; and S" D S1V-, the deviation by refraction. The deviation; By reflexion and by refraction. 15. If we make V_ V m* (2) 174 NATURAL PHILOSOPHY. Equation applicable to refraction; Equation (1) becomes sin 9 = m sin 9 (8) Equation applicable to reflexion; which answers to any refracted ray. For the reflected ray, V becomes equal to — V, and — 1 = m\ this in Equation (3) gives sin 9 = — sin

sin 9"; and if sin 9' be taken a maxi- mum, or the angle of incidence be 90°, equation (7) will give, Fig. 9. Light passing from rarer to denser medium; Angle of incidence taken 90°; m - = sin " (8) from which results a maximum limit to the angle of Maximum limit refraction. If m" be taken equal to 1,52 for the atmos- toan"leof x ' refraction; pliere and crown glass, sin 0" = 0,657, or " = 48° 15'; that is to say, the greatest angle of refraction which can exist when light passes from air into crown glass, is 41° 5' 30"; and from air into water, 4S° 15'. If the ray pass from a medium to another less dense, 180 NATURAL PHILOSOPHY. Light passing m" will be less than unity, from denser to an(j eqUal to the reciprocal rarer medium; _, of its former value ; liqua- tion (7) will then give sin 9" > sin 9' ; Angle of taking the maximum value refraction taken t . i n i 90°; for sin 9=1, we shall ob- tain from the same Equation, Fig. 10. Consequence; • / 1 sin 9 = m Trl (9) Examples; this value for the sine of the angle of incidence, which corresponds to the greatest angle of refraction when light Analogy; passes from any medium to one less dense, is the same as that found before for the greatest angle of refraction, when the incidence was taken a maximum, in the pas- sage of light from one medium to another of greater den- sity. In the case of air and glass, it is 0,657; correspond- ing to an angle of 41° 5' 30"; for air and water, the angle is 48° 15'. If the angle 9' be taken greater than that whose sine conclusion; is —T, the angle of refraction, or emergence from the m" denser medium, will be imaginary, and the light will be wholly reflected at the deviating surface. This maximum refiexio°n ^ vaiue f°r p' is called the angle of total reflexion. Light cannot, therefore, pass out of crown glass into air under a greater angle of incidence than 41 ° 5' 30 " , nor out of water into air under a greater angle than 48° 15'. § 22. The maximum limit of refraction, and the cases of total reflexion, are attended with many interesting ELEMENTS OF OPTICS. 181 results. If an eye be placed in a more refracting medium Appearances duo than the atmosphere, as that of a fish under water, it willt0 the limit of x 7 ' refraction and perceive, by the limit of refraction, all objects in the total reflexion; horizon elevated in the air, and brought within 48° 15' of the zenith, while some objects in the water would Ap- pear to occupy the belt included between this limit and the horizon by total reflexion. Those remarkable cases of mirage, where objects are seen suspended in the air, and oftentimes inverted, are Those due to explained by ordinary refraction and total reflexion. or(}m'aTy 1 t/ t/ refraction and The phenomena of mirage most frequently occur when total reflexion. there intervenes between the suspended object and spec- tator a large expanse of water or wet prairie, and towards the close of a hot and sultry day, when the air is calm, so that the different strata may arrange themselves ac- cording to their different densities. When the wind rises the phenomena cease. -If Illustration; It is well known that in the ordinary state of the at- Apparent mosphere, its density decreases as we ascend; a ray 0f efccto^thc 17 •' atmosphere on light, therefore, entering the atmosphere at S, would un-the positions of dergo a series of refractions, and reach the eye at B, with <*totw bodi^ an increased inclination to the surface of the earth ; and would appear to come from a point, Sf, in the heavens above that at S, occupied by a body from which it pro- 182 NATURAL PHILOSOPHY. ceeded. Hence, the effect of the atmosphere is to in- crease apparently the altitudes of all the heavenly bodies. Relative index § 23. Dr. Wollaston suggested a method, founded on determined by fag limit of total reflexion, to determine the relative in- dices and refractive powers of different substances. If the angle of incidence, 9', be measured by any device, Equation (9) will give, m" = sin 9 And thence the from which, Equation (7)', we find the absolute index, ' knowing that of air ; and the refractive power may then be deduced from Equation (6). Optical prism; ±N' Deviating planes and refracting angle. Deviation of a ray of light in passing through a prism; § 24. The deviating surfaces have, thus for, been supposed parallel. If they be inclined to each other, as M N, 31N', we shall have what is called an optical prism, which consists of any re- fracting substance bounded by plane surfaces intersecting each other. 31N and 31 Nr, are called the deviating planes, and the angle under which they are inclined, is called the refracting angle of the prism. § 25. To find the deviation of a ray of light in passing through a prism, let S D be the incident, D D' the first, and Df /6y/ the second re- fracted ray. The total deviation will be S'ES" ELEMENTS OF OPTICS. 183 which denote by S ; then, calling the refracting angle of the prism a, and adopting the notation of the figure, we shall have S =ED D' + ED' D = 9 - 9' + + - 4/ = 9 ++-(?'+ 4/) Equations; 180° or 77 =0 + M D D' + MD' D = a +|— 9' + ^ - 4/ or a = 4/ + 9' .....(10) Refracting angle; hence S = Cp + -^ — a.....(11) Deviation; The deviation of a ray of light in passing through a prism, is, therefore, equal to the sum of the angles of in- Eule. cidence and, emergence, diminished by the refracting angle of the prism. The refracting angle for the same prism being con-Deviation for stant, the deviation will depend upon the angles of in- fme p3r,sm x x ° depends upon. cidence and emergence. Now, from Equations (11), (10), (3), and sin -\f = m sin ^',.....(3); by a simple process of the calculus, or by trial, it may be shown, that when the angles of incidence and emer- condition for gence are equal, the deviation will be a minimum, or™1"™™ the least possible. Making 9 equal to 4^, in Equations (11) and (10), we find. 184 NATURAL PHILOSOPHY. Its use; 9 =*(* + *) 9' = 1 a which substituted in Equation (3) give Formula for refractive index. m sin \ (a + 6) m sin I a (12) we have, therefore, only to measure the deviation when a minimum, to find the index of refraction of the me- dium of which the prism is made, supposing its re- fracting angle known. This furnishes one of the best methods by which the refractive powers of different substances may be found. Application of If the substance be a liquid, we may unite two plane glasses, making any angle with each other, by means of a little cement along their edges, and place the liquid between them where it will be held in sufficient quantity by capillary attraction. the formula. incident ray § 26. "When the ray is incident at right angles upon normal to first the firg<. surface we }iaye surface; 7 " 9 = 0, 9'=0, and from Equations (10) and (11), there result, Consequences; s = ^ a, whence Final result sin (a + S) = m sin a . . . . (13) Deviation at plane surfaces by refraction, will be again referred to in a subsequent part of the text. ELEMENTS OF OPTICS. 185 §27. Let MJST, 31N', be two plane reflectors, meeting in a line projected in 31; S D, a ray incident at the point D, and con- tained in a plane perpendicular t the intersection the reflectors ; this ray will be devia- ted at the point D, of the first reflec- tor, again at the point D', of the second, and so on. Kequired the circumstances attending these deviations. Call the first angle of incidence n in which case, (/jl — n — l . i, will be negative; that is, at .the n* incidence, the ray will be on the opposite side of the The ray wm not perpendicular. It will therefore return, but not, as before, return by the _ * ' ' same path; by the same path. ELEMENTS OF OPTICS. 187 Example 3d, The angle of the reflectors being 7°, theExnniP,etWird; first angle of incidence 69°, required the number of reflexions before the ray returns, and the first angle of incidence of the returning ray. These values in Equation (15), reduce it to 0^ = 69° — n — 1 . 7° = 76° — 7° . n. If n = 10, (Pn = 76° — 70° = 6°. Suppositions; If n = 11, 0n = 76° — 77° = — 1°. Result. or the ray begins to return at the eleventh incidence and the angle of incidence is 1°. It is obvious that the angle of incidence of the return- ing ray will increase at every deviation; there will, there- fore, be some value of the increased angle which will either be equal to or greater than 90°. In the first case, 1:emarkg> the ray will be reflected by one of the reflectors into a direction parallel to the other, and in the second, this last reflexion will give the ray such a direction that it will meet the other reflector only on being produced back. § 28. Adding the first two Equations in group (14), we have 0i - 03 = 2 i, or SS'D' = 2i. That is, the angle made by the incident ray and the Angle made by the incident ray and the same ray after two reflexions; 188 NATURAL PHILOSOPHY. Equal to double same ray after two reflexions, is equal to double the an- the angle made gje 0f fae reflectors. It follows, therefore, that if the ''angle of the reflectors be increased or diminished by giv- ing motion to one of the reflectors, the angular velocity of the reflected ray will be double that of the reflector. Application of This is the principle upon which reflecting instruments tins principle. ^ faQ measureinent 0f angles are constructed. DEVIATION OF LIGHT AT SPHERICAL SURFACES. Deviation of § 29. Let 31D 0 N, be a section of a spherical surface light at spherical surfaces; separating two me- dia of different den- sities, as air and glass, having its cen- tre at C, on the line Illustration and 0 C, whicil will be called the axis of the deviating sur- face ; FD a ray of light, incident at D, Fig. 15. ,i ,» Real and virtual yond the deviating surface. In the first case, the locus foci. has but ah apparent existence, and is said to be vir- tual; in the second it actually exists, and is said to be real. The radiant is also said to be virhial when the virtual radiant wave proceeds, or, which is the same thing, the rays converge to a point before deviation. § 31. Luminous waves, like waves of sound, Acoustics Living force of § 53, become more and more diffused in proportion as intensity'of light they recede further and further from the place of primi-decreases for tive disturbance, provided their convexities continue to be increasesbfor turned to the front, and more and more concentrated converging rays. after they have been so deviated as to turn their con- cavities to the front. In other words, the living force of the wave molecules, which determines the intensity of light, will become less and less for divergent, and greater and greater for convergent rays. That portion of the living force imparted to the ethereal Living force of -.-.. t Ti.ii'1 i particles on a molecules at any one place, as a radiant, and which proceeds gplierical upon a spherical segment embraced by the bounding rays segment of a small direct pencil, can, therefore, Equations (19) and °°f™*" raec (20), be concentrated upon the ethereal molecules at another place, as a focus, by the action of a spherical devi- ating surface : and the focus, whether real or virtual, be- ^ . IT • ATld the f°<1IS comes a source of light as well as the radiant, and is as becomes a source distinctly visible. When the focus is real, the deviated oflisht wave first becomes concentrated in, and subsequently 192 NATURAL PHILOSOPHY. whence the emanates from it: when virtual, the deviated wave pro- deviated wave ' 1 proceeds for real ceeds only from the deviating surface, but with dimen- and for virtual gjons faQ same as though it had departed from the virtual foci. ° *■ focus. § 32. If the ray which is deviated at the first, be incident First deviated upon a second SUr- ra}'incident upon n -*,rr -\rr l 3 . / face 31 J\ , having a second surface; > o a radius rr, and situated at a dis- tance t, from the first, measured on the axis, we may Fig. 18 suppose this ray to have proceeded originally from Fr; and denoting the distance from the new vertex 0', to the point F", in which this ray, after deviation at the second surface, meets the axis, by f", and the index of refraction of the second medium by m', we shall have from Equation (20), Equation applicable to thc second deviation; /' m! — 1 ml r' + m'(f' + t) ' ' " (21) Fig. 19. Second deviated ray incident upon a third surface; M" X And by the same process for a third deviating surface. Equation applicable to the third deviation; /'" ' m" r" + m" {f» + ^ • (22) ELEMENTS OF OPTICS. 193 Piff. 20. M'" M" Third deviated ray incident upon a fourth surface; N'" And for a fourth, 1 (m'" - 1) J______________________ . . (23) Equation /"" ' TO!" r"' m"' {f"' + t") applicable to the fourth deviation, And so on for any number of surfaces, the law being ma- nifest. and so on. § 33. The value of f + t, deduced from Equation (20) Direct relation and substituted in Equation (21), will give a direct rela-found bctween /. . /» / the first radiant tion between/ and/, m terms of r, r, m, m and t; distance and final and the value of /" + U found from this derived equa-focal distance- tion and substituted: in Equation (22) will give a direct relation between fr" and f in terms of r, rf, r", m, m', m", t and t'; and by the same process of elimination a direct relation may be found between the radiant distance f and the final focal distance f"'—n. § 34. But in practice the distance t, is so small that it Practical relation mav, without sensible error, be neglected. Omitting t, het™en tbese • ' ' ° ° ' distances, we shall find that the first member of each of the preced- omitting t\ ino- equations becomes a factor in the last term of the second member of that which immediately follows it, and proceeding to eliminate these factors by their values, we obtain from Equations (20) and (21) 1 r mf — 1 1 | m H---;- ) m \ m! rf 13 1 1 — + m r mf . (24) Resulting equation for two surfaces, 194 NATURAL PHILOSOPHY. Eolation ( 1 between this value of -—, substituted in Equation (22), gives, conjugate focal J distances for three surfaces, omitting t\ 1 _m"-l 1 jot'-I 1 (m-1 1 \i ,9 /'" == m" r" + m"\ m! r' + m! \ m r + ~m~f) j [~0) and this value of -j^-, in Equation (23), gives, Same for four 1 W?/" — 1 surfaces, and so ~FiTu == 777 TFT i j m r on. _1 \m"-l 1 jm'-l 1 (m-1 1 \ | 1 m'"Lm'V" + m" j mV + m'l^F + ^m~f! j J ' { ' and so on for additional surfaces. Medium between § 35. If we now suppose the medium between the second and third, second and third, fourth and fifth, sixth and seventh, supposed the &c-5 deviating surfaces, the same as that in which the same as that of light moved before the first deviation, we shall have the case of a number of refracting media bounded by spherical surfaces, situated in a homogeneous medium, such as the atmosphere, for exainple, and nearly in contact. Hence, Corresponding ,1 „, 1 ,„„ 1 D values of ™. = —\ 7Tb = —- ; m'"" = —^, &C. m m m refractive indices; Resulting and the foregoing Equations reduce to jfr=(m-l).\l-^\+^.......(27) 1 m"-l 1 (____ (1 1 \ 1 ) j^, = ^^\±-^) + ^.(±.-L) + ± (29) &C, &C. ELEMENTS OF OPTICS. 195 §36. Any medium bounded by curved surfaces and Lens defined; used for the purpose of deviating light by refraction, is called a lens. Equation (27) relates, therefore, to the deviation of a small pencil of light by a single spheri- cal lens; f. denoting the distance of the radiant, and E(iuations 5 J 7 • & ' applicable to fr, that of the focus from the lens. Equation (28), re- one, two, &c. lates to the refraction or deviation by a single lens andlenses- a second medium of indefinite extent bounded on one side by a spherical surface nearly in contact with the lens. Equation (29), relates to deviation by two spheri- cal lenses close together, f and fftn denoting, as before, the radiant and focal distances. § 37. If the ravs be parallel before the first deviation, incident rays supposed / will be infinite, or 1 = 0, and Equations (20), (27), i,aralle1' (28), and (29), will reduce to yy%__J Resulting form of the preceding 5 equations; /' m? 1 m"-\ , 1 f---- (1 1\1 . ff77 = mPV" + w lm-1' W " 7'/ J ' ^,777 = m'-l -[-*- ■?*)+ m-1 • (- - -,) 5 &c, &c. The values of ff, ff1, ff'!, f,n, &c, deduced from these Principal focai Equations, are called the principal focal distances, being distanoe- the focal distances for parallel rays. Denoting these distances by Fu, FiU, Filtl, &c, and (- - -p), (p - ^777] 6zc, by —, —, -jjjy, &c, we shall have the following table, P p" P" viz. : 196 NATURAL PHILOSOPHY. Table of 1 reciprocals of Tf principal focal ' distances; Rule. F u 1< in 1 m—1 mr m—1 P m"-l + 1 (m-l\ m" \~7~) F. III! m"—l m—1 ---- + ------ P P L _ ™>""-l 1 (m"-l m-l\ F m""-l m"-l m-1 + ---77— + unit p mi P P &c, &c, &c. . (30) An examination of the alternate formulas of the above table, beginning with the second, leads to this result, viz., that the reciprocal of the principal focal distance of any combination of lenses, is equal to the sum of the re- ciprocals of the principal focal distances of the lenses taken separately; which may be expressed in a general way by the Equation, Value for the reciprocal of the principal focal distance of any combination of w^pr lenses. ^ = 2(^) (31) First members of group (30) substituted in preceding equations; :ein (-^-j, denotes the reciprocal of the principal focal distance of any one lens in the combination, the Greek letter 2, that the algebraic sum of these is to be taken, and -—, the reciprocal for the combination. Substituting the first member of the first Equation, in group (30), and the first members of the alternate Equations, beginning with the second, for their corres- ponding values in Equations (20), (27), (29), &c, we finally obtain, ELEMENTS OF OPTICS. 197 11 1 27 = -gr + ---3......(32) B«Uting J JJ m J equations for the discussion of the -*• -*-,•*• /^^ deviation of light ■F" JT f ' by one or more " lenses or by a 1 1 1 single surface. 77777 = --ft-----h -?......(3*) "TTTF/Tt = ~p-----f" ~T......(**)" J mm J Equations (33), (34), and (35), are of a convenient form for discussing the circumstances attending the deviation of light by refraction through a single lens, or a com- bination of lenses placed close together; and Equation (32), the deviation at a single surface. § 38. The several terms of these Equations are the re-To find relative ciprocals of elements involved in the discussions which measures for the vergency of are tO follow. The incident and pencil of light being Fig* 2h deviated ra^ small, the versed sine of half the arc DD', has been disregard- / ed, and the arc itself —q{-----------^- may be regarded as t/ o \ / ^-^fer----- Rays supposed to coinciding with the j^^^^^ diverge both , i • , ,i ^(^M\ before and after tangent line at the ^^ \ deviati0I1) and Vertex G, and as arc taken; naving been described about either of the points G, F', or F, as a centre, indifferently; and denoting the length of the arc 0 D by a, and the number of degrees in this arc when referred to the centre F, corresponding to the radius f by n, we shall have the proportion, 2*./: 360° :: am; whence, Number of 1 degrees in this U = * _£ arc referred to 2 if f the centre F\ 198 NATURAL PHILOSOPHY. Number of degrees in same arc referred to the centre F'; Ratio of the above values; in which *x denotes the ratio of the diameter of a circle to its circumference. When this arc a is referred to the centre F', corres- ponding to a radius ff, its number of degrees, denoted by nr, becomes, n = a . 360° _1 ~T~f 7 5 and dividing the first of these Equations by the second, we find, 1 n n' ~ / 1 /' Conclusion for diverging rays. whence we conclude that — and — measure the relative f ff divergence of the incident and deviated rays. When the devi- Conelusion for ated rays meet the axis at F', on the opposite side of the deviating surface from the radiant, the value /', being laid off in a contrary di- rection from the Fig. 22. converging rays; vertex 0, becomes negative, and the relative measure -jj-, for the convergence of these rays will be negative. Again, if the incident rays converge to a point F, before deviation, f for the same reason, would be ne- gative, and the measure for the corresponding conver- gence would be negative. And, generally, we shall find that, referring the radiant and focal distances to the ELEMENTS OF OPTICS. 199 vertex as an origin, di- vergence will be mea- sured by a positive and ^v ^^ convergence by a nega- ^ry\ tive quantity; and for .--"'s'/ \ x , J ' ,,-'' S / \ General rule for Convenience We Shall, ^-"" / J—_____1________ vergencvofrays. therefore, hereafter em- "\ K / ploy the general term ^":: ;^!\/ vergency to express either y^^\. of these conditions of the ^ rays, indifferently. § 39. The power of a lens is its greater or less capacity Power of a lens; to deviate the rays that pass through it. In Equations (33), (34), (35,) &c, 1_ _L, _L_, &c, -^ u un mm will measure the vergency of parallel rays after devia- tion ; and as these measures are expressed in functions of the indices of refraction, and _, or (------ ) &c, p \ r r I they will be constant for the same media and curvature, and may be employed as terms of comparison for the other two terms which enter into the Equations to which they respectively belong. From what has been said, it is apparent that -^,in Jo Equation (31), will measure the vergency of parallel rays after deviation by any combination of spherical lenses p0wer of a lens whatever, and will consequently be the measure of the or fombinatiou 7 x %j 0f lenses; power of the combination; and as f__J, is the measure of the power of any one lens of the combination, we have this rule for finding the power of any system of lenses, viz.: Find the p>ower of each lens separately, and take the ^uIe- algebraic sum of the whole. § 40. It will be convenient to express the relation in Equations (32), (33), (34), &c, by referring to the centre 200 NATURAL PHILOSOPHY. To find a relation of curvature of the deviating surfaces as an origin. For between the this purpose, let 0 D conjugate focal L x distances when be a section of the deviating surface, and denote the dis- tances of the radiant and focal points from the centre C, by c and cr, respectively; we have by inspection, the centre of curvature is taken as the origin; Fijr. 24 Substitutions and reductions; / = r + c, f=r+c', which in Equation (19), give, after reduction, Relation for one surface; For a second surface; 1 m — 1 m — =---- + — ore (36) and for a second deviating surface whose centre of curva- ture is at a distance t, from that of the first, we ob- tain from Equation (36), m! — 1 c tt r + m c'-t . . . . (37) and for a third, whose centre is at a distance t', from that of the second, For a third surface; m tf ju tt + m a c"-t ...... (38) Relations for a lens, &c. c' being eliminated between Equations (36) and (37), a relation between c and c", will result; in like manner, c" being made to disappear by means of this derived equation and Equation (38), there will result an equa- tion in terms of c"' and c, and so for others. 41. Eetaining the thickness t, of the medium between ELEMENTS OF OPTICS. 201 the two deviating surfaces to which Equations (19) and Retaining t, for (21) relate, we obtain from the first by adding t, to both members and reducing to a common denominator, /.,,,__ mrf + (m— 1 .f + r) t^ m — 1 . f + r and this substituted in Eq. (21), at the same time making And ml = suppose these surfaces to , which is supposing the ray to pass into the bound a medium ''v immersed in first medium after having traversed the medium bounded another; by the two deviating surfaces, that Equation reduces to, J. 1 — m m(m — 1 .f + r) it r Final relation for --- • • (ot/) a single lens mrf+ (f-m--1 + r) t retaining t. which gives a direct relation between the conjugate focal distances in the case of light deviated by a single lens. APPLICATION OF THE PRECEDING THEORY TO THE DEVI- ATION OF LIGHT BY REFRACTION THROUGH THE VARI- OUS KINDS OF SPHERICAL LENSES. § 42. A lens has been defined to be, any medium Application of bounded by curved surfaces, used for the purpose of l^^'f^ deviating light by refraction; the surfaces are generally various spherical spherical. lemes. A, called a double convex lens, is bounded by two spherical sur- faces, having their cen- tres and the surfaces to whicli they corres- pond, on opposite sides of the lens. When the Fig. 25. Geometrical representations of the spherical lenses. 202 NATURAL PHILOSOPHY. lens; Plano-convex; Double convex curvature of the two surfaces is the same, the lens is said to be equally convex. B, is a lens with one of its laces plane, the other spherical, this latter face and its cen- tre being on opposite sides of the lens, and is called a plano-convex lens. Double concave; C5 js a ct-ovble concave lens ; each curved face and its centre lying on the same side of the lens. D, is & plano-concave lens, having one face plane and the other concave. E, has one face concave and the other convex, the con- vex face having the greater curvature; this lens is called a meniscus. F, like the meniscus, has one face concave and the other convex, but the concave face has the greater curvature ; this is called a concavo-convex lens. The line containing the centres of the spherical surfaces, is called the axis. Plano-concave Meniscus; Concavo-convex. Different cases § 43. A moment's consideration will show that all the ^^J^du; circumstances of vergency attending the deviation of light by any one of these lenses, will be made known by Equation (33), it being only necessary to note the dif- ferent cases arising out of the various combinations of surfaces by which the lenses are formed; these cases de- pend upon the signs of the radii. Equations (33), (34), (35), &c., were deduced on the Rule for signs of supposition that r is positive, the concave side of the surface being turned towards incident light; it will of course, § 29, be negative when the convex side is turn- ed in the same direction. Besides, / was taken positive for a real radiant, or when the rays are supposed to di- verge from any point upon the axis of the lens, before deviation ; on the contrary, it will become negative when 999999999 ELEMENTS OF OPTICS. 203 the rays are received by the deviating surface in a state And of conjugate r» . • j ^ ^ » t ,t ^ rT^^ f°cal distances. ot convergence to a point behind the lens. The signs of /', ff', etc., will be positive when the deviated rays meet the axis on being produced back. The foci are then virtual. When the rays meet the axis on the opposite side of the lens or lenses, ff, f", &c, become negative, and will correspond to real foci. The several lenses may be characterized as follows : 1 Double Convex, %\ Piano-Convex, convex side to in- cident light, Do. plane side to inci- — r and + rr — r and + r'- + r = oo and + r' oo Z\ dent light, . . ' Meniscus, convex side turned to incident light, . . Same, concave side do. do. 4 Double Concave,..... 'Piano-Concave, concave side to incident light, . \Same, plane side to do. do.-}-r? — oo, and—r' Concavo-Convex, concavo side to incident light, ^ <^ rpf^ + r, + r' ^ Same, reversed, . . r > r'', — r, — r' h< 6^ r < r', — r, — rf r > r, + r, + r' + r, — r' + r,+rr oc Characteristics (A) °^ ^ie vari°us lenses. § 44. To discuss the properties of any one of these Discussion of the properties of any lenses, resume lens Equation (33), de- FI* 26- termine the sign of 4r> by refe' rence to its gen- eral value in Equations (30), and the table above, and then proceed to make various suppositions in regard to the position of the radiant and deduce the corresponding places of the focus. 204 NATURAL PHILOSOPHY. Double con/ex § 45. As an example, let us take the double convex Ions taken as an i . lens. example; Equation (33), is General equation; / tt 1_ 1 F.. f II Value for reciprocal of principal focal distance; and, Equation (30), and Table (A), F. a m — 1 P — __m {-+-)■ \ r r I and as long as m > 1, we shall have, Equation for discussion; l 1 tt F- + f n (40) Fig. 27. Real radiants between For ___ > —, or / > Fn, f" will be negative, and the vergency of the deviated rays will be ne- pnncipai focus gative. That __ ^-p, ] \ \ and infinity; is to Say, if a F wave proceed from a point Give real foci, upon the axis in front of the lens between the limits Fu, the principal focus, and infinity, it will be concentrated after deviation, into a point upon the same line behind, and the focus will be real. For F < a T or Fig. 28. f /1 \ cT^es 208 NATURAL PHILOSOPHY. Covex lenses collect, and concave lenses disperse the light. A similar table may also be constructed by formula (34), for a combination of any of the spherical lenses taken two and two, and by formula (35), for any com- bination taken three and three, and so on. In general, it may be inferred from the preceding table, that convex lenses tend to collect the incident rays, while concave lenses, on the contrary, tend to scatter them. To construct the focus; § 47. Transposing, in Equation (33), —? to the first member, we get / tt f Fu Illustration; Interpretation. Fig. 31. J5T- which shows that the vergency after, diminished by that before deviation, gives a constant vergency measured by the power of the lens. Hence, to construct the focus, draw the extreme ray FD, and from the point D, the line D H, mak- ing with the incident nxjFD, produced, the angle IID K, equal to the power of the lens; D H will be the de- viated ray, and the point F", where it meets the axis, will be the focus, For, in the triangle F D F", the angle D F" 0, measured by _, diminished hjDF F", measured by — , is equal to HD K, measured by__-; which i the geometric interpretation of the above equation. is 48. Suppose the conjugate foci to be in motion, Conjugate foci supposed in j i motion; an(l denote any two consecutive values of / by x and x, ELEMENTS OF OPTICS. 209 and the corresponding values of f" hjy and?/', then station and Equation (33), equations; + y f„ x 111 = -TS- + Transformations y fu ' x' ' subtracting the second from the first we find, 1111 y yr g> g>' and reductions; reducing to a common denominator, and writing for the products y y' and x xr, the quantities/"2 and/2, to which they will be sensibly equal, the Equation becomes y' — y x' — x , and dividing by the interval of time t, during which Time t, the change from x to xr takes place, which is the samelntroducod^ as that from y to y', we have y' — y 1 x' — x 1 /"2 t f or, V" V L_ _ J__-......(41)cc f,2 -f 2 ' v ' distances and Relation between conjugate focal velocities of f"2 f in which V denotes the velocity of the radiant, and V" that of its conjugate focus; and since the denomina- tors must always be positive, being squares, the signs of the two velocities must always be alike. Whence we conclude, that in lenses a change in the place of the radiant will always be accompanied by a change of its conjugate in the same direction, and that the rate of change in the one will be to that of the other . ... „ Conclude, that in as the squares of their respective distances from thelensesconjugate lens directly. This has an important application in thefocialwar action of lenses when employed to form images. direction. 14 m s move in same 210 NzVTURAL PHILOSOPHY. If the lens be a sphere. § 49. If the lens be a sphere, mf = — > t = 0, and —-> from Equation (36), being substituted in Equation (37), we obtain c n 2 (m - 1) 1 --------H— mr c (42) 50. If in Equation (20), we make r infinite, we get Deviation at a plane surface by refraction; or /' mf ™>f = A which answers to the case of a small pencil deviated at a plane surface separating two media of different densities, as air and water. On the supposition that the denser milium- radiant is in the denser medium, m becomes —, and ' m this in the preceding Equation gives /= mf] that is, to an eye situated without this medium, the dis- tance of the radiant Fif 3^ from the deviating sur- face will appear dimin- ished in the ratio of unity to the relative in- dex of refraction of the ray in passing from the • denser to the rarer me- dium. This accounts for the apparent eleva- tion above their true positions of all bodies beneath the surface of fluids, as the bottom of a vessel partly filled with water, and the apparent bending of a straight stick at the surface when partly immersed in the same fluid. Illustration; Appearances accounted for. ELEMENTS OF OPTICS. 211 APPLICATION TO THE DEVIATION OF LIGHT BY SPHERICAL REFLECTORS. §51. In reflexion, we have only to consider one de-E(luation . . tit • applicable to a viating surface. Equation (20) applies here by making spherical concave m = - 1, which reduces it tO, reflector; r _1 7 (43) But two cases can arise, and these are distinguished bv the sign of the radius. The reflector may be concave towards incident light, in which case r will be positive, or it may be convex towards the same direction, when r will be negative. Equation (43) relates to the first case, which will now be discussed. If the incident rays be parallel, — = 0, and / Incident rays parallel; /' 2_ r or. r f = -■- = Ft 2 Fiff. 33. ,— Principal focal distance; Hence the principal focal distance is equal to half radius, and Equation (43), reduces to /' 1 F 1_ f /,n Equation for ^ / discussion; Now, this Equation is only concerned with the re- flected wave, and if this wave be concentrated at all after deviation, it must be upon that part of the axis on the side of the incident light, and hence /', for a 212 NATURAL PHILOSOPHY. Real radiants reaj focus muS)t be positive, and for a virtual focus ne- beyond the x ' principal focus; gatlVC 1 1 As long as -_ > ._ , or / > Ft, ff will be positive, A / and thc vergency of the deviated rays will be positive; that is, a wave proceeding from a point in front of the reflector between the principal focus and infinity will, after deviation, be concentrated into some other point in front after reflexion. When __ < -_, or/< Ft,f will be negative, and A / principal focus; tke vergenCy wiH be negative; in other words, a wave proceeding from a point on the axis between the vertex and principal focus, will never be concentrated after de- viation, but will appear to proceed from a virtual focus behind. If the radiant be at the centre of curvature, / = 2 Ft, and Real radiants within the Radiant at the centre of curvature. Real radiants beyond the centre; f' = 2Fl = r; that is, a wave proceeding from the centre of curvature will, after deviation, return to that point. For f> 2Ft orf>r', we have 1 1 w ^ 7->O?0r/ Give real foci or the focus will be between the reflector and centre, and between the 111 11 centre and since — _ < we find - < _L, or /' > F.\ 80 principal focus; J?, / JO, f F that the focus will be found between the centre and prin- cipal focus. , ELEMENTS OP OPTICS. 213 If Real radiants between the centre and f r; beyond the J •*• i centre; that is, the focus will be at a greater distance from the reflector than the centre. When f = F,we shall have _ = 0 ; that is, the ver-Real radiant at .. t.i J the principal gency will be zero, which shows that a spherical wave focus. proceeding from the principal focus will be transformed by deviation into a plane wave, whicli can only be con- centrated at a distance /' = oo . If the vergency before incidence be negative, / will be negative, and Equation (44), becomes Jl_ 1 1 ft ' jp \ /• •••••• \xoj Virtual radiants, Hence, /' will always be positive, and the ver^encv „ , • o j Always give real positive; that is, when a wave is proceeding to con- foci. centration in a point behind a concave reflector, it will after deviation, be concentrated into some other point in front. Equations (44) and (45), show that -__, which mea- sures the vergency of deviated rays, is always algebrai- ^XTtors n iii l-i. t t analogous to cally greater than _, which measures the vergency of convex lenses. the incident rays. Hence, concave reflectors, like con- vex lenses, tend to collect the rays of light which are deviated by them. 214 NATURAL PHILOSOPHY. Different cases § 52. By discussing the several cases that will arise in attributing different signs to r and f and various values to the latter, we shall find the results in the following TABLE. _ ,, . Reflector Incident pencil. Table for convex A * and concave 1 7 Sign of -j, reflectors: ^ r- ri Concavo Diverging Reflect, pencil -n ,------A-----N +/ f 5 A___L i \{f>F' \{ + } |Conver< t F, f J \ W/ HDs ;es ergres f 1 l j Converging ) <---1--- 11 -/ i i^. / \AA\ Converges more. f{m+fs \ \~-j\ Convex ■ — F < \r~f\ I Diverges more. />*,}{ - } J Diverges. f 1 1 ) - (Converging <-----b — > f ^ w \S + M Converge U -/ I F, f S I " / \ \ less. Conclusions. from which we perceive that convex reflectors tend to scatter the rays and concave reflectors to collect them. 53. If —, in Equation (44), be transferred to the first member, we find 1_ /' / 1_ F. Sum of the vergencies after which shows that the vergency after, increased by that and before i r» -1 . ,. deviation belore deviation, is a constant vergency, which is mea- constant; sured by the power of the reflector; and to construct ELEMENTS OF OPTICS. 215 the focus, draw the Fi& 34 extreme ray FD, and the line D F', mak- ing with the incident ray the angle FDF' equal to the power of the reflector, the point F', where this line meets the axis, will be the focus. The reason is obvious. Construction of foci for reflectors. § 54. By a process entirely similar to that of § 47, we For reflectors may find from Equation (44), which appertains equally to conJu«ate f0C1 ** X \ J i XX X *J mnvpin nnnn a concave or convex reflector by assigning to —r its pro- F. move in opposite directions. per sign, V'_ A2 V f (46) and because V and "Fhave contrary signs, we conclude that the conjugate foci in the case of spherical reflectors proceed, when in motion, in opposite directions. § 55. Equation (43), by making r infinite, reduces to JL - _JL ff ""-"/ Deviation by reflexion at plane surfaces; or. "Which shows, that in all cases of deviation of a pencil by a plane reflector, the divergence or convergence will not be altered ; and if the rays diverge before deviation, they will appear after deviation to proceed from a point conclusion. as far behind the reflector as the real radiant is in front; but if they converge before deviation, they will be brought to a focus as far in front as the virtual radi- ant is behind the reflector. 216 NATURAL PHILOSOPHY. SPHERICAL ABERRATION, CAUSTICS, AND ASTIGMATISM. Spherical aberration; Incident pencil not small; Illustration; Longitudinal aberration; Lateral aberration; § 56. Thus fixr the discussion has been conducted upon the supposition that the pencil is veryvsmall, and that 2, the versed-sine of the angle d, included between the axis and the radius drawn to the j^oint of incidence of the extreme rays of the pencil, is so small, that all the products of which it is a factor may be neglected. If, however, 2 be retained, and Equation (18) be solved with reference to ff, the value of this latter quantity will be expressed in terms of m, f, r and 2, and may be written /,' = *;; (47) Fig. 85. and if the semi-arc of the deviating surface, denoted by 0, and of which 2 is the versed- sine, be made to vary from zero to any magni- tude sufficient to embrace the ex- terior rays of any definite pencil, it is obvious that //, must have an infinite number of values, and that each value will give the focus for those ravs only which make up the surface of a cone and are incident at equal distances from the vertex. This wan- dering of the deviated ravs from a single focus is called aberration, and when caused by a spherical deviating surface, as it is in the case under consideration and in practice generally, it is called spherical aberration. When estimated in the direction of the axis, it is called longitu- dinal, and at right angles to the axis, lateral aberration. If we represent the second member of Equation (19) by M, that Equation may be written f' = M (19)' • ELEMENTS OF OPTICS. 217 and subtracting this from Equation (47), we find Measure of longitudinal and lateral // —/' = Mz -- 31.....(48) Serration, and their laws of variation; in which the first member denotes the length of the por- tion F' Fz, of the axis along which the different foci will be distributed, and will measure the longitudinal aber- ration. The lateral aberration is measured by the length of the line F' L, drawn through the focus of the rays near the axis of the pencil and perpendicular to the axis of the deviating surface. The linear length of the arc, 0 D — r. d, is called the radius of aperture, and it is found Radius of that in all cases of ordinary practice, the longitudinalapeitme- aberration varies as the square, and the lateral aber- ration as the cube of the radius of aperture. If in Equation (48), we make m = - 1, we shall have Aberration for a •tit. n reflector. the longitudinal aberration for a spherical reflector. If the value of fzf in Equation (47), be substituted for / in Equation (18), and we write /" for /', then solve the equation with reference to /", still retaining z, and /Y, n tt Aberration for a take the difference between this value of / and thatiens. given by Equation (27), we shall find the longitudinal aberration for a single lens; and that for any number of lenses placed close together might be found by the same process. § 57. We perceive that a spherical wave of any con- siderable extent deviated at a spherical surface, will not, , , -, .n ,. General effects of m general, be concentrated at, nor will it appear to pro- SI,ilt.ricai ceed from, the same point; but if we conceive the wave aberration; to be divided into an indefinite number of elementary zones by planes perpendicular to the axis of the devi- ating surface, each zone will have its particular point of concentration or of diffusion, according as the foci are real or virtual. Moreover, longitudinal aberration di- minishes the focal distance, that is, in general, // is less than f\ and the deviated rays which are in the same .Effc?J0' . J ' ^ longitudinal plane and on the same side of the axis, will intersect aberration; 21S NATURAL PHILOSOPHY. each other before they do this latter line. Thus, if FD Fig. 36. Geometrical illustration; Explanation of the figure; Caustic curve; Caustic surface; Section of the deviated wave by a plane through the axis of the surface; When the caustic will be real. be the exterior, and FDr its consecutive incident ray, D Fz and D' F", the corresponding deviated rays, these latter will intersect each other at some point as cf, on the same side of the axis OF; in like manner, if D"F' be the next consecutive deviated ray to D' F", it will intersect this atter in same point as c", and so for other deviated rays up to that one whicli coincides with the axis. The locus of these intersections c!, c", &c, is called a caustic curve / and if the curve be revolved about the axis 0 F, it will generate a caustic surface. This surface will spring from the focus of the axial rays at F', as a vertex, and open out into a trumpet-shaped tube towards the deviating surface. The deviated wave will no longer be spherical, but will be of such shape that its section df d" d"' o', by a plane through the axis of the deviating surface, will be the in- volute of the section c' c" F', by the same plane, of the caustic surface, taken as an evolute. If after deviation the wave approach the caustic, the latter will be real, Flg'37* being formed by the doubling over, as it were, of the deviated wave up- on itself, thus pro- ducing at the cusp cf double the ethe- real agitation due ELEMENTS OF OPTICS. 219 to either segment Fz c' or c' cJ separately. If, on the con- virtual caustic. trary, the wave recede from the caustic on being devia- ted, the caustic will be virtual. Caustics are finely illus- trated on the surface of milk when the light is reflected upon it from the interior edge of the vessel in which Illustration- it is contained. § 58. We have only spoken of a pencil of light whose 0blique pcncil. radiant is on the axis, which is usually called a direct pencil. When the radiant is off the axis, the axial ray of the pencil becomes oblique to the deviating surface, and the pencil is said to be oblique. In the case of an oblique pencil, however small, the deviated rays will not, in general, meet the axis as in the case of the direct pencil, but will all intersect two lines at right angles to each other and not situated in the same plane. These lines are called focal lines, and the property of the de-Focai lines; viated rays by which all of them intersect both of these lines, is called astigmatism. Astigmatism. § 59. It is, generally, not possible to deviate a spherical Aberration wave of sensible magnitude by a single lens or surface of spherical form without aberration, and yet the practi- cal difficulties in grinding lenses and reflectors to any other figure render it necessary to adhere to this shape. Fortunately, however, two or more lenses may be so united that the aberration of one shall counteract that of another, and light may thus be deviated without aberration. When such combinations are used, a wave proceeding from one point may be made by deviation to proceed from, or concentrate in, some other point. Such points are called apianatic foci, and the combi-Apianatic foci, nations wicli produce them are said to be apianatic. combinations. » 220 NATURAL PHILOSOPHY. OBLIQUE PENCIL THROUGH THE OPTICAL CENTRE. Oblique pencil through the optical centre; Explanation. To find the § 60. We have seen, article (19), that a ray undergoes no ultimate deviation when it passes through a medium bounded by two parallel planes. If, then, in the case of an oblique pencil the rays diverge sufficiently to cover the entire face of a lens, there may always be found one at least which will enter and leave the lens at points where tangent planes to its surfaces are pa- rallel. This ray being taken as the axis of a very small pencil proceeding from the assumed radiant, will con- tain the focus of the others, the distance of which from the lens, in very moderate obliquities, will be measured by f", given in Equation (27). This is obvious from the fact that in the immediate vicinity of the tangen- tial points the surfaces, which are spherical, will be symmetrical in respect to the line which joins them. To find where the ray referred to intersects the axis of the lens after deviation at the first surface, let M jfNr M' repre- sent a section of a concavo- Fie. 38. optical centre of COnvex lens, of which the ra- a lens or a . dms CO of the first surface is surface Relation from figure; r, and C Of of the second is rf; SP and S' P' the traces of two parallel tangent planes. Denote by t the distance 0 0\ between the surfaces measur- ed on the axis, and by e the distance OK, from the first surface to the intersection of the line joining the tangen- tial points P, P\ with the axis. Then, since the radii C P and C P', drawn to the tangential points, must be parallel, the similar triangles CP JTand C P! K, will give the relation, CO __ CO1 CK~~ C K ELEMENTS OF OPTICS. 221 and replacing these quantities by their values, j T r Same in other == ~~t 7 terms; — e r — t — e from which we find r t r e = (49) r' — r r' r KesuIt defined. L0 But this value of e is constant, whence we infer that all rays which emerge from a lens parallel to their di-Optical centre rections before entering it, proceed after deviation at tin first surface in directions having a common point on the axis. This point is called the optical centime, and may lie between the surfaces or not, depending upon the figure of the lens. If we suppose but one deviating surface, then the medium behind must be of indefinite extent, in which case r and t will become infinite and sensibly equal, and Equation (49) reduces to e = r. That is to say, the optical centre of a single deviating optical centre of surface is at the centre of curvature. a single surface; If the lens be double concave, the radius r' becomes negative, and the value of e, in Equation (49), becomes r t e = — -7-t—> r + r and if the faces be equally concave, r will equal /, and t e = ~2 Of a double That is, the optical centre is midway between the faces.concavelens' 222 NATURAL PHILOSOPHY. Of a double convex lens; If the lens be double and equally convex, r becomes negative, and the result will be the same as above. In the case of a meniscus with its concave face turned towards incident light, the radii will both be positive, and r > r, whence Of a meniscus; rt (p____rp Of a plano-convex lens. In a plano-convex lens having its plane face turned towards incident light, r will be infinite, and r' finite and positive, and e = — t. which brings the optical centre to the vertex of the curved face. The student may determine in the same way the optical centre of the other lenses. OPTICAL IMAGES. Optical images; Explanatory remarks; § 61. The surface of every luminous body is made up of a vast number of radiants, from each of which waves of light proceed in all directions. These waves cross each other; and if any deviating surface be presented, it be- comes the common base of a multitude of pencils, whose vertices are the radiants which make up the surface of the body. Some one ray of each of these pencils will pass through the optical centre of the lens, and those rays in the immediate vicinity of this one constituting a small pencil will be brought to a focus upon it as an axis, and hence for each radiant in the surface of the body there will be a corresponding conjugate. These conjugate foci make up a second luminous surface, from which waves will pro- ceed as from the original body; and this surface is called ELEMENTS OF OPTICS. 223 an image of the body, because to an eye so situated as to Image of a bodyi receive these new waves, the object, though often modi- fied in shape and size, will seem to occupy the position of the new surface. An optical image is, therefore, an assemblage of con- optical image jugate foci answering to a series of radiants on thedefilied; surface of some object; and its formation consists, in so deviating portions of the waves of light which proceed its formation from the object, as either to concentrate them in someconsistsin; new positions from which they may proceed as from the object itself, or to cause them to move from these new positions without having at any time occupied them. In the first case the image will be real and in the second Eeaiimage; virtual. In general, but a part of each wave can be de- viated by the use of spherical deviating surfaces to sat- isfy these conditions, for those portions remote from the Yirtual undeviated ray of each pencil cannot, in consequence of aberration and astigmatism, be brought to accurate ver- gency. imago, §62. To ascertain the relation between an object and To find the its image, let us suppose the deviation to be produced relat1ion bet*c.f ° J x x x an object and its by a lens, so thin that its thickness may be neglected, image formed by whicli is the usual case in practice. The optical centre alens; G, may be taken 4-U • • x» Fig. 39. as the origin ot co- ordinates. Denot- ing by I, the dis- tance from this point to any as- sumed point P in the object, and writing this quantity for /, in Equation (33), which we may do without sensible error, we get / n__ F. // F 1 + ~f \o\Jj corresponding to an assumed radiant point. 22-4 NATURAL PHILOSOPHY. section of the Let the object be a plane, perpendicular to the axis object assumed to Qf ^ leng ^ ^^ WJU be a rf ht Hne p ✓) Q^ . be a right line; ' ° ^ ' the angle included between the axis of any oblique pen- cil and the axis of the lens. When the pencil becomes direct, 61 will be zero, and I will equal f. But, generally, we have General relation; 1 = cosd ' this in Equation (50), reduces it to Equation of the image of a right line; /" = F. II i j___a cos 6 1 + / (51) which is the polar equation of the image referred to the optical centre as a pole. It is the same in form as the polar equation of a conic section, which is Is thc same in form as that of a conic tection; r — A(l-e2) 1 + e cos v conclusion; Whence we conclude that the image of a straight line perpendicular to the axis of the lens which forms it, is a conic section, and comparing the two Equations, we find', /" = r, Equations compared; Ftt = A(l — c>) = B3 (52) e = 7' (53) 8 = v. ELEMENTS OF OPTICS. 225 For the same lens, Fn is constant: its value —, in E1, F„ = GO , Conditions for the different conic sections. or according as the distance of the object is infinite; greater than the principal focal distance of the lens; equal to this distance ; less than this distance; or zero. If the section P Q be supposed to revolve about the axis of the lens, it will generate a plane, and the image a curved surface whose nature will depend upon the dis- tance of the object. We have seen that a positive value for ffr, answers sign of the focai . . . t t ,. li-^ii? distance for a to a virtual, and a negative value to a real focus; lens? will indicate so, if the points of the image be indicated by positive whether the values for/", the image will be virtual; if by nega-^°^s tive values, real. For a concave lens, Fu is positive, and Equation (51), answers to this case. For a convex lens, Fu is negative, and Equation (51), becomes 15 226 NATURAL PHILOSOPHY. Image will be real for a convex lens as long as the object is beyond the principal focus; /"=- F. II F (54) // / cos 6 and the image will always be real as long as F. f u. cos 6 < 1, or -I-i>F«> cos 6 Illustration. Fig. 40. That is, if-from the optical centre, with &■ radius equal to the principal focal distance, we describe the arc of a circle, and this arc cut the object, the image of all that part of the object in- cluded between the points of intersection A and A! will be vir- tual, while that of the parts without these lim- its will be real; if the distance of the object exceed that of the prin- cipal focus, the whole image will be real. § 63. Multiplying both members of Equation (51), by sin 0, it becomes Equation (51) transformed; r^& = ^^l .... (55) i jp COS 0 " and giving to 0, its greatest value for any assumed object, /tan 6 will be the length of that portion of the object on ELEMENTS OF OPTICS. 227 the positive side of the axis as long as d is positive and less Explanation of than 90° ; /" sin 0, is the distance of the extreme limit 0fterms; the image of this portion of the object from the axis; and writing / tan o* = 8^ Substitutions; /"«**« = *„, Equation (55) becomes, after dividing both members by / tan d, *// Fi, -----• Equation (55) J , 777 transformed; + F, cos d " If the linear dimensions of the object be small as com- pared with its distance from the optical centre, we may compared with write unity for cos b, the image will, § 48, and Eq. (52), its distance from sensibly coincide with Sn, and the above equation °P ] reduces to $.. F. u -*■ a f + K (56). In which the essential signs of all the quantities correspond to a concave lens. For a convex lens, Flt is negative, and Equation (56) becomes **// -*// (K^7\ Equation for a T— -- — "^----TT • • • • \0 i J. $ f___ Jf convex lens; Equations (51) and (56), show that the image of every real object formed by a concave lens is virtual, erect, and less than the object, while Equations (54) and (57), show that the image of every real object formed by a convex lens is real as long as the object is beyond the .. , .., Images formed principal focus, is inverted, and less or greater than the by concave and object, depending upon the distance of the latter fr0m conyexlcilses- r>>,Q NATURAL PHILOSOPHY. when the image the optical centre. When the distance of the object is the obje^T twice that of the principal focus, Equation (57) becomes i. = -i. other cases. and the object and image are equal in size. When the object is within the principal focus, it is less, and when beyond the principal focus greater, than its image. iMation between § 64. If we make 0 equal to nothing in Equation (51), linear dimensions ntt ^ coincide with the ax;s of t]ie J ^ } {h of the object and ,y 7 ° image; will measure the distance of the image from the opti- cal centre, while / will measure that of the object on the same line. Denoting these distances by Dn and D, respectively, substituting them in Equation (51), clear- ing the fraction in the second member, and dividing both members by D, we find J>„ .. F, D f + F/ which, in Equation (56), gives 8.. D II ___ // S D . . . . (58) snme in words. That is to say, the corresponding linear dimensions of an object and of its image are to each other directly as their respective distances from the optical centre. imnge formed by § 65. If an image be formed by deviation at a sin- devmtion at a jQ surface j|-s p0jnts will be readily found by means of single surface; ° x j j Equation (36); the optical centre, in this case, being at the centre of curvature. Writing f for c, and ff for c\ that Equation becomes ELEMENTS OF OPTICS. 229 1 m—1 m making f = oo, 1 m—1 1 /' r f; ' ' " Equation applicable; (KCk\ Principal focal ' distance; hence, 1 1 m % Equation for ■ft ' Jp * J* ' discussion; or, Same in another form; r_ fF _ __F__ f+mF, 1,nLF ' f For an oblique pencil passing through the optical cen- tre, we have, on the supposition that the object is a right line perpendicular to the axis of the surface, -ft_ Ft (CKCW Same for an J — "^VW «... \0\J) oblique pencil 1 -f_____L COS 0 through the J optical centre. wherein---- = —, as in article (62). f v § G6. If the image be formed by reflexion, m = — 1, and Equation (60) becomes JO . /r*-<\ Image formed by — jp~ ~ • • • • ^J-; reflexion; 1 + jLu COS 6 f 230 NATURAL PHILOSOPHY. illustration; since for a concave re- flector, Ft, Eq. (59), becomes negative. This is a polar equation of a conic section, the nature of which will result from the relation of Ft to /. It will, § 62, be an ellipse, parabola, or hyperbola, according as Fi- 41. %y ^W,y f>Fr,f = F,;oYf^ h 'ject is to the corresponding dimension of the image, as the distance of the object from the centre is to that of the image from the same point. And a moment's reflec- tion will show us that all real images must be in front, while all virtual images must be behind the reflector. § 68. We get the point in which the image cuts the axis by making Equation for discussing a concave reflector; or 6 =0. /'=- F. 1+5 / / I— . . . (62) interpretation of This value of /' being negative, the image will be found on the left of the centre, the distance f having been taken positive to the right. As long as f is posi- tive, the image will lie between the centre and reflec- tor, f' will be less than f and the image, consequently, less than the object. When f is zero, ff will also equal zero, and the object and image will be equal and occupy ELEMENTS OF OPTICS. 231 the centre. When f becomes negative, or the obiectPositionsand relative size of passes between the centre and reflector, ff will be posi- the image when tive as long as f < F, and the image will pass without, theobJectis ' between the ff will be greater than/, or the image will be greater than centre and the the object. When/, being still negative, is equal to Ft vertex* or the object is in the principal focus, the image will be infinitely distant. The object still approaching the reflec- tor, / will be greater than Ft; / becomes negative again and the image will approach the reflector from behind it, and will be greater than the object till/ = 2 Ft or the object be in contact with the reflector, when/' will equal /, and the image and object be of the same size. 8 69. When the reflector is convex, r is negative, the Convex reflector; principal focal distance Ft Equation (59), is positive, and Equation (60) becomes Ft /go\ Equation J jp * * " * Vu / applicable; 1----^ cos 6 f and maMng d = 0, // . J # # # (64:). Equation for ■f discussion; F'1 This value of/' is always positive, greater than F4, and Ee]ations less than 2 Ft, for all values of/ between 2 Ft and in- between the finity, or for any position of the object from the surface fof/^jectsT of the reflector to a point infinitely distant in front. In the latter position, /' is equal to Ft, or the image is in the principal focus. It follows also, that the image, which will always be virtual for real objects, will be elliptical, erect, and smaller than the object. § 70. If we make / positive, greater than Ft and less same for virtual than 2 Ft, the object will be virtual; the image real, objects* erect, and greater than the object. 232 NATURAL PHILOSOPHY. OF THE EYE AND OF VISION. The eye; § 71. The eye is a collection of refractive media which concentrate the waves of light proceeding from every point of an external object, on a tissue of delicate nerves, called the retina, there forming an image, from whicli, by some process unknown, our perception of the object arises. These media are contained in a globular en- Four coatings velope composed of four coatings, two of which, verv envelop the , ° refractive media; unequal in extent, make up the external enclosure of the eye, the others serving as lining to the larger of these two. Fig. 42. Graphic representation of the eye; The cornea; shape of the eye; The shape of the eye is spherical except immediately in front, where it projects beyond the spherical form, as indicated at d e dft, which represents a section of the human eye through the axis by a horizontal plane. This part is called the cornea, and is in shape a segment of an ellipsoid of revolution about its transverse axis which coincides with the axis of the eve, and which has to the conjugate axis, the ratio of 1,3. It is a strong, horny, and delicately transparent coat. Immediately behind the cornea, and in contact with ELEMENTS OF OPTICS. 233 it, is the first refractive medium, called the aqueous Aqueous humour, which is found to consist of nearly pure wa- ter, holding a little muriate of soda and gelatine in solution, with a very slight quantity of albumen. Its refractive index is found to be very nearly the same as that of water, viz. : 1,336, and parallel rays having the direction of the axis of the eye will, in consequence of the figure of the cornea, after deviation at the surface of this humour, converge accurately to a single point. At the posterior surface of the chamber A, in con- tact with the aqueous humour, is the iris, ^^,Iris; which is a circular opaque diaphragm, consisting of muscular fibres by whose contraction or expansion an aperture in the centre, called the pupil, is diminished PuPil; or increased according to the supply of light. The ob- ject of the pupil seems to be, to moderate the illumi- nation of the image on the retina. The iris is seen through the cornea, and gives the eye its color. In a small transparent bag or capsule, immediately behind the iris and in contact with it, closing up the pupil, and thereby completing the chamber of the aque- ous, lies the crystalline humour, B; it is a double con- crystalline -• ,i , p ,i , humour; vex lens of unequal curvature, that of the anterior sur- face being least; its density towards the axis is found to be greater than at the edge, which corrects the spherical aberration that would otherwise exist; its mean refractive index is 1,381, and it contains a much greater portion of albumen and gelatine than the other humours. The posterior chamber 0, of the eye, is filled with the vitreous humour, whose composition and specific yitr, gravity differ but little from the aqueous. Its refractive v index is 1,339. At the final focus for parallel rays deviated by these hu- mours, and constituting the posterior surface of the cbam"Ret.na. ber C, is the retina, h h h, which is a net-work of nerves of exceeding delicacy, all proceeding from one great 0pticnerve. branch 0, called the optic nerve, that enters the eye IVollS humour; 231 NATURAL PHILOSOPHY. Fig. 42. Graphic representation of the eye; <7——. if o Choroid coat; Sclerotic coat Inverted images formed on the retina: obliquely on the side of the axis towards the nose. The retina lines the whole of the chamber C, as far as ii, where the capsule of the crystalline commences. Just behind the retina is the choroid coat, JcJc, cov- ered with a very black velvety pigment, upon which the nerves of the retina rest. The office of this pig- ment appears to be to absorb the light which enters the eye as soon as it has excited the retina, thus prevent- ing internal reflexion and consequent confusion of vision. The next and last in order is the sclerotic coat, which is a thick, tough envelope d d' d", uniting with the cor- nea at d d", and constituting what is called the white of the eye. It is to this coating that the muscles are attached which give motion to the whole body of the eye. From the description of the eye, and what is said in arti- cle (62), it is obvious that inverted images of external objects are formed on the retina. This may easily be seen by removing the posterior coating of the eye of any re- cently killed animal and exposing the retina and cho- roid coating from behind. The distinctness of these im- ages, and consequently of our perceptions of the objects from whicli they arise, must depend upon the distance ELEMENTS OF OPTICS. 235 of the retina from the crystalline lens. The habitual Habitual position position of the retina, in a perfect eye, is nearly at the ° focus for parallel rays deviated by all the humours, be- cause the diameter of the pupil is so small compared with the distance of objects at which we ordinarily look, that the rays constituting each of the pencils employed in the formation of the internal images may be regarded as parallel. But we see objects distinctly at the distance of a few inches, and as the focal length of a system of lenses, such as those of the eye, Equation (25), increases with the diminution of the distance of the radiant or obiect, it is certain that the eye must possess the power „ J J *■ x Lye possesses the of self-adjustment, by which either the retina may be power of made to recede from the crystalline humour and theself"adjustment: eye lengthen in the direction of the axis, or the curva- ture of the lenses themselves altered so as to give greater convergency to the rays. The precise mode of this ad- justment does not seem to be understood. There is a limit, however, with regard to distance, within which vision becomes indistinct; this limit is usually set down at six indies, though it varies with different eyes. The Limit of distinct limit here referred to is an immediate consequence of vision; the relation between the • focal distances expressed in Equation (25), for when the radiant or object is brought within a few inches, the corresponding conjugate or im- age is thrown behind the point to which the retina may be brought by the adjusting power of the eye. With age the cornea loses a portion of its convexity, the power of the eye is, in consequence, diminished, and Long sightedness distinct images are no longer formed on the retina, the ° ° , and its remedy; rays tending to a focus behind it. Persons possessing such eyes are said to be long sighted, because they see objects better at a distance ; and this defect is remedied by convex glasses, which restore the lost power, and with it, distinct vision. The opposite defect arising from too great convexity in the cornea is also very common, particularly in young persons. The power of the eye being too great, the 236 NATURAL PHILOSOPHY. Images on the retina are inverted; But objects appear erect: shortsightedness image is formed in the vitreous humour in front of the and its remedy. retina^ an(j t]1Q remec|y is in the use of concave glasses. Cases are said to have occurred, however, in which the prominence of the cornea was so great as to render the convenient application of this remedy impossible, and relief was found in the removal of the crystalline lens, a process common in cases of cataract, where the crys- talline loses its transparency and obstructs the free pas- sage of light to the retina. The fact that inverted images are formed upon the retina, and we, nevertheless, see objects erect, has given rise to a good deal of discussion. Without attempting to go behind the retina to ascertain what passes there, it is believed that the solution of the difficulty is found in this simple statement, viz.: that we look at the object, not at the image. This supposes that every point in an image on the retina, produces, without reference to its neighboring points, the sensation of the existence of the corresponding point in the object, the position of which the mind locates somewhere in the axis of the pencil of rays of which this point is the vertex; all the axes cross at the optical centre of the eye, which is just within the pupil, and although the lowest point of an object will, in consequence, agitate by its waves the highest Explanation of point of the retina affected, and the highest point of the above. the object the lowest of the retina, yet the sensations be- ing referred back along the axes, the points will appear in their true positions and the object to whicli they belong erect. In short, instead of the mind contemplat- ing the relative positions of the points in the image, the image is the exciting cause that brings the mind to the contemplation of the points in the object. Base of the optic It may be proper to remark here, that the base of to^nS6nSible ^e °P^C nerve5 where it enters the eye, is totally insen- sible to the stimulus of light, and the reason assigned for this is, that at this point the nerve is not yet divided into those very minute fibres which are capable of being affected by this delicate agent. ELEMENTS OF OPTICS. 237 2. The apparent magnitude of an object is deter- APParent magnitude of an mined by the extent of retina covered by its image. object determined; Fig. 48. If, therefore, i? Rf be a section of the retina, by a plane through the optical centre C, of the eye, and A B = I, ab~\ the linear dimensions of an object and its image in the same plane, we shall have, from the similar triangles CAB and Cab, X= — Qa . — s (65)? a \ Dimension of image of an object on the retina; denoting by s, the distance of the object. And for any other object whose linear dimension is V and distance si3 calling the corresponding dimension of the image \, x = — C a . —, 6 Same for a second object; and since C a is constant, or very nearly so, X : x •: — : —, ' s s I Proportion; that is, the apparent linear dimensions of objects are as their real dimensions directly, and distances from the 1 , ., ^ Rule first; eye inversely. But _, may be taken as the measure of s the an^le B C A =b Ca, which is called the visual an- \ 23S NATURAL PHILOSOPHY. Rule second; gle, and hence the apyparent linear magnitudes of objects are said to be directly proportional to their visual angles. Small and large objects may, therefore, be made to ap- pear of equal dimensions by a proper adjustment of their distances from the eye. For example, if X = Xy, we have Example for illustration; I V ■ V or. V .6 I Numerical data; and if I = 1000 feet, s = 20000, and V = 0,1 of a foot, or little more than an inch, Result 20000_.^l=2f 1000 ' the distance of the small object at which its apparent magnitude will be as great as that of an object ten thou- sand times larger, at the distance of 20000 feet. MICROSCOPES AND TELESCOPES. Microscopes; Explanatory remarks; § 73. From what has just been said, it would appear that there is no limit beyond which an object may not be magnified by diminishing its distance from the optical centre of the eye. But when an object passes within the limit of distinct vision, what is gained in its apparent in- crease of size, is lost in the confusion with which it is seen. If, however, while the object is too near to be distinctly visible, some refractive medium be interposed to assist the eye in bending the rays to foci upon its retina, distinct vision will be restored, and the magnifying process may ELEMENTS OF OPTICS. 239 Its operation illustrated; be continued. Such a medium is called a single micros- single t tt 'ii*i i • • -i r» i microscope cope, and usually consists of a lens, whose principal focal distance is negative and numerically less than the limit of distinct vision. To illustrate Fi&-44- the operation of this instrument, let MN be a section of a dou- ble convex lens whose optical centre is 0 / Q P an object in front and at a distance from C equal to the principal focal distance of the lens; E the optical centre of the eye, at any distance behind the lens. Th6 rays Q 0 and P C, containing the optical centre, will undergo no deviation, and all the rays proceeding from the points Q and P, will be respectively parallel to these rays after passing the lens; some rays, as W-#Explanatj011 of from Q, and ME from P, will pass through the optical the figure; centre of the eye, and will belong to two beams of light whose boundaries will be determined by the pupil, and whose foci will be at q and p on the retina, giving the visual angle, MEN = PC Q; Relation from same; or the apparent magnitude of the object P Q, the same as if the optical centre of the eye were at that of the lens. And this will always be the case when an object occupies the principal focus of a lens whatever the dis- tance of the eye, provided the latter be within the field of the rays. Without the lens, the visual angle is QEP < P CQ; Effect of the hence, the apparent magnitude of the object will be in- ^scope; creased by the lens. Calling X and X, the apparent magnitudes of the ob- ject as seen with, and without the lens, we shall have, \ 240 M:i;rmtudes of NATURAL PHILOSOPHY. PQ P_Q„ _1_ . 1_ anobjecUvith '<'•'' (f Q '' £ Q' ' Q Q ' F Q and without the lens compared; or, by using the notation employed in Equation (33), and calling E Q, the limit of distinct vision, unity, X \ F„ (w-l)(±+*) • • (66) when the lens As long as F < 1, or the principal focal length of maybe used as a j ^ ^ j.^ Qf djstinct yision the an- single ' x microscope; parent size of the object will be increased, and the lens may be used as a single microscope. We can now understand what is meant by the power of a lens or combination of lenses, referred to at the close What is meant i by the of article (39). ---, which was there said to measure magnifying -^ u power of a single q^q power of a lens, we see from Equation (G6), expresses microscope; x t . the apparent magnitude of an object referred to that at the limit of distinct vision, taken as unity; and what- ever has been demonstrated of the powers of lenses gen- erally, is truetof magnifying powers. Thus, in Equation (31), we have the magnifying power of any combination of lenses equal to the algebraic sum of the magnifying powers taken separately. Should any of the individuals of the combination be concave, they will enter with signs contrary to those of the opposite curvature. Rule for The power of a single microscope is, Equation (66), ^wl-rofa"sin-yie equal t° the limit of distinct vision divided by its princi- microscope; pai focal distance, and the numerical value of the power will be greater as the rfractive index and curvature are greater. § 74. To obtain a general expression for the visual an- gle under which the image of an object formed by a lens, and having any position in reference to the eye, ELEMENTS OF OPTICS. 241 Fig. 45. is seen, let Q P, be an object in front of a concave lens. From P, draw through the optical centre E, the line P E; from P, draw the extreme ray P M, and from i/draw M S, making with P Mproduced the angle SMT equal to the power, of the lens; then will, § 47, MS be the corresponding deviated ray, and its intersection p, with the ray P E, through the optical centre, will be a point in the image ; from p, draw p q, parallel to P Q, till it is cut by the ray Q E, through the other extreme of the object and optical centre; p q will be the image. Let 0, be the optical centre of the eye; then denoting the visual angle p Oq by A, we have, To find the visual angle under which an image formed by a lens is seen; / A = _ 2P qp Oq Eq-OE Value of visual angle; and representing the distances Q E by f Eq by f", and EO by d, we find, qp = QP.£l- Eq-EO=f"-d; and hence A- Qp f" ■ Same in other terms; and denoting the visual angle PEQ by A!, r A A' f'-d - d 16 Ratio of visual (67) angles with and without the lens, 242 NATURAL PHILOSOPHY. Sign of this ratio depends upon; Eye placed so as to see the image formed by a concave lens; Fig. 46. The angles A and A' will have contrary signs when on opposite sides of the axis of the deviating surface. The relation expressed by this equation answers to a concave lens in which f" will, Equation (27), be posi- tive for a real object. Moreover, d is positive, the eye being on the same side of the lens as the object; but that the image may be seen the eye must be on the opposite side, in which case d will be negative, and the Equation becomes Equation corresponding to this case; A A' 1 + i /" (68) Real image formed by a convex lens; whence we conclude that objects will always appear dimin- ished when seen through concave lenses. If the lens be convex and the object be situa- ted beyond its principal focus f" will be nega- tive, and Equa- tion (68) becomes Equation corresponding; A_ A' 1- f" (69) Distinct vision supposed possible and supposing distinct vision possible for all positions for all positions „ _ . Jr of the eye. ot the eye, it appears, ELEMENTS OF OPTICS. 243 1st. That when the object is at a distance from the conclusion first; lens greater than that of the principal focus, in which case there will be a real image, the lens will make no difference in the apparent magnitude of the object, provided the eye is situated at a distance from the lens equal to twice that of the image. 2d. At all positions for the eye between this limit second; and the image, the apparent magnitude of the object is increased bv the lens. 3d. At a position half way between this limit and Third; the lens, the apparent magnitude of the object would be infinite. 4th. The eye being placed at a distance greater than Fourth. twice that of the image, the apparent magnitude of the object will be diminished by the lens. 5th. When the distance of the object from the lens_ . . Fifth- is equal to that of the principal focus, in which case f" becomes infinite, the apparent magnitude will be the same as though the eye were situated at the optical centre of the lens, no matter what its actual distance behind the lens. image formed by a reflector is seen; § 75. The visual angle under which the image formed to find the visual by a reflector is seen, is found in the same way. Thus, let anglG under J ? u which an in P Q be an object in front of a convex re- flector M N. From the extreme point P, of the object, and through the optical centre C, draw the ray P C\ from the same point P, draw to the extreme of the. reflector the ray P M, and from M draw MS, making with P M, the angle P MS equal to the power of the Explanation reflector; JLT^will, § 53, be the deviated ray, and its intersection with P C, will give the image of the point 244 NATURAL PHILOSOPHY. construction of P. Draw p q. paral- the image formed by a reflector; Fig. 18. lei to P Q, till it is intersected by Q C, drawn through the op- posite extreme of the object and optical centre, and we have the image. Let the optical centre of the eye be at 0; then, de- denoting the visual angle p 0 q by A, will Value of visual angle with the reflector; Ratio of visual angles with and without the reflector. A IF ®p °g- Oq' OQ' Oq' and representing, as before, C Q, Cq, and C 0, by /, P 0 f\ and d, respectively, and the visual angle —J^byA', CQ we have A f A' d-r (70) We shall not stop to discuss this Equation. In practice all positions of the eye; § 76. We have supposed; in the preceding discussion, distinct vision is distinct vision to be possible for all positions of the eye ; not possible for but this we know depends upon the state of convergence or divergence of the rays. If, however, the image, when one is formed, instead of being seen by the naked eye, be viewed by the aid of another lens, so placed that the rays composing each pencil proceeding from the object shall, after the second deviation, be parallel, or within such limits of vergency that the eye can command them, the object will always be seen distinctly, And the image is and either larger or smaller than it would appear to therefore viewed the miassisteci eye depending upon the magnitude of through an eye J ' L ox <~> lens. the image, and the power of the lens used to view it. ELEMENTS OF OPTICS. 245 As most eyes see distinctly with plane waves or parallel Position of the rays, this second lens is usually so placed that the imageeye lens* shall occupy its principal focus; and where this is the case, we have seen that the apparent magnitude of the image will be the same as though the eye were at its optical centre. Fig. 49. Refracting telescope; The of the lens image p q, being in the principal focus m n, draw from the point p, the line construction for one pencil of p 0, to the optical centre of this lens; the rays from p li€rht traced will, § 73, be deviated parallel to this line, and the line the retina- 0' K, througli the optical centre 0' of the eye, paral- rel to p 0, will determine by its intersection K, with the retina, the place upon that membrane of the image of the point P. Calling the principal focal distance of this lens, (Fd) ; d, in Equation (67), will equal f" + (F„), and that equa- tion will become, by first making f" and d negative and then replacing d by this value, to A_ A' f tt (FJ General equation ( i 1) made applicable to this telescope; and if the object P Q, be so distant that the rays com- posing each of the small pencils whose common base is MN, may be regarded as parallel, f" becomes Fn, and we have, A_ A' ZL t*;,) Eatio of visual (72) angles for parallel rays; 246 NATURAL PHILOSOPHY. Fisr. 49. Refracting telescope; Compound microscope; Field and eye lenses; Rule for magnifying power. Objects appear inverted. Galilean telescope; Construction of image on the retina; Equation (71) involves the principles of the compound refracting microscope, and refracting telescope y and Equation (72), which is a particular case of (71), relates to the astronomical refracting telescope. The lens MN, next the object, is called the object or field lens, and mn, the eye lens. The magnifying power in the first case, is equal to the distance of the image from the field lens divided by the yirincipal focal length of the eye lens; and in the second, to the principal focal length of the field lens, divided by that of the eye lens. The ratio of A to A!, being negative, shows that ob- jects appear inverted through these instruments, the vis- ual angles of corresponding parts of the object and im- age being on opposite sides of the axis. § 77. If instead of a convex, a concave lens be used for the eye lens, the combination will be of the form used by Galileo, who invented this instrument in 1609. In this construction, the eye lens is placed in front of the image at a distance equal to that of its principal focus, so that the rays composing each pencil shall emerge from it parallel. Draw through the point p, where the image of P would be formed, the line p 0, to the optical centre 0 of the eye lens, and through the optical centre 0' of the eye, the line 0' JTparallel top 0, its intersection K, with the retina will give the image of the point P on the back part of the eye. ELEMENTS OF OPTICS. 247 Fig. 50. Galilean telescope; The rule for finding the magnifying power of this Magnifying stru have, instrument is the same as in the former case: for we pov™r fTd ' analytically; d=f"-{Fly, which in Equation (67), after making/*", and d, negative. gives A 11- A (FJ' (73) Ratio °f Tisual angles; and for parallel rays, A_ A' F. II m . . . (74) Same for parallel rays; The second member being positive, shows that objects objects appear i Pi*Pf*f seen through the Galilean telescope appear erect. § 78. If we divide both numerator and denominator of Equation (72), by F„. (F„), it becomes, 1 A A' iF) 1 F, II Magnifying power in terms of the powers of the lenses; and denoting by Z, the power of the field, and by Z, that of the eye lens, we have A_ A' L (75) Ratio of visual angles; 248 NATURAL PHILOSOPHY. Rule for magnifying power. that is, the magnifying power of the astronomical tele- scope is equal to the quotient arising from dividing the power of the eye lens by that of the field lens. Fig. 51. Geometrical illustration of the — field of view; Q General explanation; § 79. If E, be the optical centre of the field, and 0 that of the eye lens of an astronomical telescope, the line E 0, passing through the points E and 0, is called the axis of the instrument. Let Qr Pr be any object whose centre is in this axis, and qf p' its image. Now, in order that all points in the object may appear equally bright, it is obvious from the figure, that the lens must be large enough to embrace as many rays from the points Pf and Q', as from the intermediate points. It is not so in the figure; a portion, if not all the rays from those points will be excluded from the eye, and the object, in consequence, appear less luminous about the exterior than towards the centre, the brightness increasing to a certain boundary, within which all points will appear Field of view; equally bright. The angle subtended at the centre of the field lens, by the greatest line that can be drawn within this boundary, is called the field of view. To find this angle, draw m JV and Mn to the opposite extremes of the Determined by lenses, intersecting the image i\\p and q, and the axis in X; then will p q be the extent of the image of which all the parts will appear equally bright. Draw qEQ and p E P\ the angle PE} Q— pEq,\s the field of view, which will be denoted by I; construction; First form of its value; 23 r (76) ELEMENTS OF OPTICS. 249 but pq m n ~XO .Xr Transformations; (77) to find X 0 and Xi; call the diameter MNof the object lens «, that of the eye lens /3, and we have a:/3 ::EX:XO cc + (3:(3::EX+XO:XO Proportions; hence, XO=P(f"+(FJ); a+ p and in the same manner, EX= " .(/" +TO); Relations from the figure; a Xr=f"-EX=f'- —(f'+iF,)) = ^f^; these values in Equation (77), give pq = Pf"-* TO /" + TO : Substitutions; and this in Equation (76), gives, by introducing the powers of the lenses, T (3 I — a L Ju . —-----=— 1 + £ (78) Final value for field of view. The rays of each of the several pencils emerging from the eye lens parallel, will be in condition to afford dis- 250 NATURAL PHILOSOPHY. Fte. 51. Geometrical illustration; Proper position for the eye indicated in telescopes; tinct vision, and the extreme rays m Or and n 0', will be received by an eye whose optical centre is situated at 0'. If the eye be at a greater or less distance than 0', from the eye lens, these rays will be excluded, and the field of view will be contracted by an improper posi- tion of the eye. It is on this account that the tube con- taining the eye lens of a telescope usually projects a short distance behind to indicate the proper position for the eye. From the similar triangles p Oq and m Orn, we have Distance of optical centre of the eye from that of the eye lens ; Position of the eye for the Galilean telescope ; Arrangement for changing the distance between the lenses. 0 0'=^.rO=?J£+4.{F,) .... (79) pq pi — a I, This also applies to the Galilean instrument, by chang- ing the sign of I, which will render 0 0', negative. The eye should, therefore, be in front of the eye-glass in order that it may not, by its position, diminish the field of view; but as this is impossible, the closer it is placed to the eye-glass the better. When the telescope is directed to objects at different distances, the position of the image, Equation (27), will vary, and the distance between the lenses must also be changed. This is accomplished by means of two tubes which move freely one within the other, the larger usu- ally supporting the object and the smaller the eye lens. Terrestrial telescope; § 80. The terrestrial telescope is a common astronomi- cal telescope with the addition of Avhat is termed an ELEMENTS OF OPTICS. 251 erecting piece, which consists of -a tube supporting at Erecting piece; each end a convex lens. The length of this piece should be such as to preserve entire the field of view, and its position so adjusted that the image formed by the object glass, shall occupy the principal focus of the first lens of the erecting piece, as indicated in the figure, Terrestrial telescope; . formed; in which case a second image will be formed in the prin- cipal focus of the second lens of the erecting piece, and the corresponding linear dimensions of these images will be to each other as their distances from the lenses whose Eelationbetwcen principal foci they occupy, Equation (72). These images the two images bein^ viewed through the same eve lens, viz.: that or the telescope, their apparent, will be directly as their real magnitudes. Hence, denoting by A and A" the visual angles subtended at the optical centre of the eye lens by the first and second images respectively; by V and I" the powers of the first and second lenses of the erecting piece, we have, 41 A F en V F 'ff 9 ei Ratio of their visual angles at the optical centre of the eye lens ; in which Ftl and Ftl„ are the principal focal lengths of the first and second lenses. Multiplying this by Equation (75), member by member, we have for the magnifying power of the terrestrial telescope, A' L I" Magnifying power of the terrestrial telescope. 252 NATURAL PHILOSOPHY. objects appear And since the ratio of A" to A' is positive, objects will appear through this instrument erect. Compound microscope; § 81. If, now, the object approach the field lens, f", in Equation (71), will increase, and the magnifying power become proportionably greater; but this would require the tube containing the eye lens to be drawn out to ob- tain distinct vision, and to an extent much beyond the limits of convenience if the object were very near. This difficulty is avoided by increasing the power of the ob- ject lens, as is obvious from Equation (54); and when this is carried to the extent required for very great prox- imity, the instrument becomes a compound microscope, which is employed to examine minute objects. The com- Compound microscope; Fig. 53. same in principle pound microscope not differing in principle from the as refracting i -i ., .„ . x r telescope; telescope, its magnifying power is given by the Equa- lis magnifying power; tion, 4.= -Jl = m r and substituting for _ its value in Equation (40), we have Same in a different form; A_ A' m'l- T' / F„ ELEMENTS OF OPTICS. 253 or, writing D for _; and representing, as before, the / powers of the field and eye lenses by L and I, A A' D- V Final value for magnifying power; from which it is obvious that the magnifying power may be varied to any extent by properly regulating the po- May be varied; sition of the object; but a change in the position of the object would require a change in the position of the eye- glass, and two adjustments would, therefore, be neces- sary, which would be inconvenient. For this reason, it is usual to leave the distance between the lenses unal- tered and to vary only the distance of the object to suit distinct vision. It is, however, convenient to have the power of changing the distance between the glasses, as by that a choice of magnifying powers between certain limits may be obtained, and for this purpose the object glassesin and eye glasses are set in different tubes. different tubes. Usual practice; Fig. 54. Reflecting telescope; § 82. If the field lens of the astronomical telescope be replaced by a field reflector MX, whose optical centre is at C, as indicated in the figure, we have the common astronomical reflecting telescope. C being the optical centre, d becomes equal toff - (F„), and Equation (70), becomes, by first changing the sign of /', and then sub- Explanation; stituting this value for d, 254 NATURAL PHILOSOPHY. Magnifying power for terrestrial objects; Same for celestial objects. A_ A' (FJ and for plane waves or parallel rays, A_ A' F. i . . the dimensions of be to those ot the image, as their respective distances the object and from the lens B; if, therefore, the lens B be mountedimage; in a tube which admits of a free motion in that con- taining the lens A, its distance from the figure may be varied at pleasure, and the image on the screen made larger or smaller, the instrument, at the same time, be- ing so moved as to keep the screen in the conjugate corresponding to the focus occupied by the glass slide. The instrument with an arrangement by which this can be accomplished, is called the phantasmagoria. In or- phantasmagoria. der, however, that the deception may be complete, there must be some device to regulate the light, so that the illumination of the image may be increased with its increase of size, not diminished, as it would be without such contrivance. SOLAR MICROSCOPE. § 90. This is the same as the magic lantern, except solar that the light of the sun is used instead of that from E 266 Solar microscope; NATURAL PHILOSOPHY. a lamp. D E, is a long reflector on the outside of a window shutter, in which there is a hole occupied by the tube containing the lenses. Fig. 64. Essential parts and manner of usinjc. The object to be exhibited is placed near the focus of the illuminating lens A, so as to be perfectly en- lightened and not burnt, which would be the case were it at the focus. CHROMATICS. Chromatics; Color in light corresponds to pitch and harmony in sound; Explanatory remarks; § 91. Chromatics is a name given to that branch of optics which treats of Color. Color is to light what pitch and harmony are to sound. We have seen, in Acoustics, that by the principle of the coexistence and superposition of small motions, any number of sonorous waves may exist at the same time and place, and pro- duce, through the organs of hearing, an impression dif- ferent from that produced by either of the waves when acting singly. The united tones proceeding from the various voices of a full choir of music, for example, im- press the ear very differently from the insulated note of the acute treble, the medium tenor, or the full, deep-toned bass; and as each voice is partially or wholly suppressed in succession from a full strain of concordant sounds, while others are reinforced, the mind ELEMENTS OF OPTICS. 267 marks the change, and attributes to it a distinct and Analogy between specific character. the action of c 'ii i • sonorous and bo it is with the luminous waves which act upon the luminous waves; organs of sight. These come to us from the sun, and other self-luminous bodies, of every variety of length ca- pable of affecting the eye; they coexist and are super- posed upon the retina, and by their united influence give us the impression of white light; and when one white light after another of these waves is enfeebled, while others produced: are strengthened, each new combination gives us a dif- ferent impression, and each impression we call a color. The longest waves capable of affecting the eye corres- pond to red, and the shortest to violet or lavender grey. But how are individual waves either suppressed or separated from the group which produce the sensation Principles which of white light? The answer is, by the principles of m-pr°CUC€ terference and of unequal refrangibility. COLOR BY INTERFERENCE. Colors of Gratings. Colors of gratings; §92. Recalling the expla- nation of § 7, let MX be a wave front proceeding from a source 0. Assume any point Or, in front of the wave, and draw the straight line Or 0. Take the dis- tance A B equal to half the length \, of the longest, and A B' equal to half the length \, of the shortest wave capable of affecting the organs of sight; and make B C = CD = AB — \\. construction of With (9r as a centre, and the radii 0TB', OrB, OrC\*s™' jr. Fig. 60. 0 JS ^\ M J? \ j ^^ N / ^ \/ c \ y JS & =^> S^c ^v ^>^/y JL W 2 £ 268 NATURAL PHILOSOPHY. Colors of gratings; Fig. 6a Construction of figure; 2 £ Even numbered portions of main wave opposed to the odd portions. Consequence of stopping the even portions; Effect of longest waves most increased at 0,.; Effect of shortest waves most increased at 0 • OrD, &c, describe arcs cutting the wave front in V, b, c, d, &c.; then will the portions Ab, be, c d, &c, in the immediate vicinity of A, partially, and those remote from the same point, wholly, interfere, § 7, and neutralize each other's ef- fects at Or; for, at this point the secondary waves from the successive points of the portion A b, beginning at A, will be opposed to those from the corresponding points in the portion b c, beginning at b, being in opposite phases ; and it is -plain that if the several portions be numbered in order from A, that those distinguished by the even will be opposed to those designated by the odd numbers, the odd portions tending to displace the molecule at Or in one direction, and the even ones in the opposite direction. Now, conceive the even portions be, de, &c, to be stopped by the interposition of some opaque screen; the odd portions no longer being neutralized, will have full effect upon Or, which will become greatly more lumi- nous by the conspiring action of the longest waves— that is, by the waves whose length is Xr. But the shortest secondary waves proceeding to Or from the portion b b', on one end of A b, will interfere and neutralize those from an equal portion A a, at the other end, so that the effect of the longest waves at Or, will be increased in a much greater proportion than that of the shortest. From the construction Or c — Or A = \. Take a point Ov, such that Ov c - Ov A = \, then will the point Ov, for the same reasons, receive the greatest possible dis- turbance from the action of the shortest waves which are here in the same phase, while the effect of the longest waves, at this point, will be less than at Or, being no longer in the same phase. The effect of the waves whose ELEMENTS OP OPTICS. 269 lengths are intermediate between Xr and X^ will have their preponderance upon molecules between these two points, and the space Or Ov, should exhibit correspond- ing effects. And this is found by experiment to be the case. For when a grating is formed by fine parallel Effects of furrows i • r. n n , - .t — Fy-----5 red and violet colors of the wth fringe from the n . d . X„ centre. s In which, because Xr is greater than \, xr, which de- notes the distance from Y to the red color of the nth fringe, will be greater than xv, which represents the dis- tance of the corresponding violet color from the same point, Subtracting the second from the first, we get The colors separate from each other; Xt_xv = "L^L(K--kv) .... (89) From which we see that the different colors will sep- arate more and more as the fringe to which they belong And the dark recedes from the centre Y. The black intervals will, ^™ls **** disappear. therefore, be encroached upon, and at no great distance from Y will disappear. To find the order of the last insulated fringe, denote by a ,i / i i\j.i n ii t» • To find the order xn+l and #nthe distances of the {n+l)th and nth fringes of thela9t from Y' and by Xr the length of the wave for red, then insulated fringe; will Equation (88) give 278 NATURAL PHILOSOPHY. Notation and equations; Xn+l = (n +1). Xr. d ®n = S n .X, . d S whence, taking the difference, we obtain for the inter- val between the reds of two consecutive fringes, Interval between the reds of two consecutive fringes; Xlvx.\ — Xn — Xr . d ~~S~ and placing the second member of this Equation and that of Equation (89) equal, we find n *d . (X - x) = x' • d S S whence Order of the last insulated fringe. n — Experimental illustration. Experiment performed in vacuum; \~\ (90) That is to say, the order of the last insulated fringe is denoted by the number of units in the quotient arising from dividing the length of the red wave by the dif- ference of the lengths of the red and violet waves. This result is beautifully illustrated by interposing between the screen and grating some medium which will arrest all the waves but those which correspond to a particular color. When this is done, the fringes will be greatly multiplied in number beyond that of the nth order determined by Equation (90). § 98. Thus far the waves have been supposed to pro- ceed, after passing the grating, in the atmosphere. But when the experiment is performed in vacuum, with the same grating and same position of the screen, the fringes are found to dilate and separate from each other; when per- formed in a medium of greater density than the air, as ELEMENTS OF OPTICS. 279 in water or glass, the fringes are reduced in width and Experiments crowded towards the centre ; and what is remarkable, fd^nirmedium and important to observe, this latter effect is found, bythanair; careful experiments, to be exactly proportional to the in- dex of refraction of the medium as referred to that of atmospheric a ir. ISow, referring to Equation (88), it is easy to see that this change in the position and width of the fringesCauseofthe can only arise from a change in X, which denotes the position and length of the waves, since S and d are, bv the conditions width of the irin^es * of the experiment, constant; and from the relations of x and X, in that Equation, it follows that the length of luminous waves of the same color are shorter in propor- tion as the indexes of refraction of the media in which they exist are greater. But, Equation (2), these indexes Lengths of waves vary inversely as the velocities of wave propagation, and in diffcrent " . . media; hence the lengths of the waves are directly proportional to the velocities with which they are transmitted through different media. The cause by which the lengths of the waves are thus altered in the direct proportion to their Principleofwavo acceleration and velocities, is called the principle of wave acceleration retardation. and retardation. § 99. Returning to the experiment in air; if a very thin plate of glass be interposed in front of one of the interposing a grate openings, and parallel to the plane of the grating, Pla^of glass the whole system of fringes will be shifted towards conditions. the side of the interposed glass. If an exactly similar plate be placed in front of the other opening, and parallel to the first plate, the fringes will be re- stored to their original position. If one of the plates be slightly inclined, so as to cause the waves passing through it to traverse a greater thickness, the fringes will all move towards that side, and by gradually in- creasing the inclination, they will pass entirely out of sight. Taking plates of any other medium, possessing a greater refractive index than glass, and of the same 280 NATURAL PHILOSOPHY. Effect of interposing a plate of any medium. Effect on the lengths of the rays; This last effect investigated; Illustration; Explanation; Fig. 72. ^ C thickness as before, it is found that the effects just no- ticed will be increased, and in the direct ratio of the refractive indexes of the media. In the shifting of the fringes, it is evident that the lengths of the rays which correspond to the central one are made unequal, and that the differences as to lengths existing among the rays which appertain to the other fringes, are not the same as before the interposition of the medium. We will now investigate this change. For this purpose, let the waves from both openings pass througli a prism of any medium, as glass, hav- ing a very small refracting angle, i, the first face being held pa- rallel to the plane of the grating. The thickness of the prism tra- versed by two interfering waves will be different; call this diffe- rence, which is r n in the figure, d. Draw nn', parallel to KL\ with 0, as a centre and 0 r as a radius, describe the arc rr'. It is obvious that the number of waves in the length A n + Or will be equal to the number in the length Cn'+Or', since the circumstances are the same in both routes ; the only difference, if there be any, must lie in the paths n r and n' rf. Since the angle made by the rays A 0 and CO, is very small, these rays will enter the first sur- face under very small angles of incidence, and both be- ing refracted towards the perpendicular, their direction through the prism will be nearly normal to that surface; hence, denoting by b, the distance r nr, we have Equation from figure; d = b . sin i; but under the above supposition, the angle of incidence at the second surface will be equal to i; and denoting ELEMENTS OF OPTICS. 281 the corresponding angle of refraction by r diminishes. the central illumination. The reverse effect will arise either on increasing the size of the body, or diminish- ing its distance from the screen. § 102. If a portion of the pencil be transmitted througli Pencil admitted a small and well defined circular aperture and received circul°raperUira upon a screen, concentric rings will also be produced; and if the transmitted portion be viewed through a con- vex lens, the hole will appear as a bright spot, encircled by rings of the most vivid colors, whicli undergo a great varietv of changes, both as regards tint and linear dimen- sions in varying the distance of the lens from the aper- 284 NATURAL PHILOSOPHY. tare, and that of the aperture from the radiant or lumi- nous point. Light When the light is transmitted through two very small throu™?^ apertures, close together, rings corresponding to each circular apertures wm \>q formed as before, and in addition there will be found a number of straight parallel fringes between the centres of the circles, and at right angles to the line joining them; two other sets of parallel fringes will also be seen in the form of St. Andrew's cross proceeding from the space between the centres; and by multiplying the number of the apertures and varying their relative dimensions, a set of phenomena arise of exceeding bril- liancy and beauty. n ,.. 8 103. The colored fringes of shadows and small aper- colored fringes of tures, as well as all appearances referred to under this lh^t°uwa.and ^ead, are cause(* by interference; the interference taking place between the secondary waves from the edges of the body or aperture and those from that portion of the primitive wave whicli is not intercepted. Names originally These phenomena were at one time called the i? flection applied to them. or ^graci{on 0f light, and were supposed to arise from some peculiar action exerted by the edges of bodies on the rays as they passed near them. Effect of If the refractive index of the medium in which the increasing the experiments are performed be increased, the phenomena refracting index r x , of tbe medium, indicate a diminution in the lengths of the waves in the same ratio. COLORS OF THIN PLATES. au media § 104. Transparent, and indeed all media, when re- exiiiwt colors duced to very thin films, are found to exhibit colors thin films; which vary with the thickness of the film. These are called the colors of thin plates, and the easiest way to exhibit them is by means of a soap bubble blown from the end of a quill or the bowl of a common smoking ELEMENTS OF OPTICS. 285 Thin plate of air formed; pipe. As the bubble increases in diameter, and the Familiar n't i • t ! . .i.i . t , i exhibition of the fluid envelope is reduced in thickness at the top by colors of thia gradual subsidence toward the bottom, many colored plates; and concentric rings will be seen around the point of least thickness. At this point, the color will be found to change, first appearing white, then passing through blue to perfect blackness, the rings the while dilating till the bubble is destroyed. The same is true of any other medium, whether gase- ous, liquid, or solid. These different colors being exhibited upon the same when the plate plate of variable thickness, no single color can be iden- 1^i0ck^(!rm titled with its chemical composition. When of uniform thickness, a single color only will be seen, and this will change as the thickness of the plate changes. A thin plate is very con- venientlv formed of air; and for this purpose, let A B, be a plano-convex, and CD a plano-concave lens, placed one upon the other, as represented in the figure. "When this arrangement is viewed on either of the plane faces by reflected light, colors will be seen in the form of con- Appearances by centric circles about the point of contact, which, should reliected Hsht; the pressure be sufficient, will be totally black. If view- ed by transmitted light, rings whose colors when united with those of the first, form white light, and which By transmitted colors are, therefore, said to be complementary, will ap-llght; pear about the central spot, which will now be per- fectly white. With waves of a single length, as yellow, these rings are alternately bright and dark, begin- ning with the central spot; and by reflected light, dark and bright. They are broadest and have the great- est diameter in the red, and narrowest with least diam- eter in the violet; the breadths and diameters in the Effljcta dae to other colors being intermediate and varying in magni-waves of a single hide in the order of the spectrum from red to violet. Itlength; is by the superposition of these rings, or the waves 2S6 NATURAL PHILOSOPHY. Newton's scale; which produce them, that the different colors appear in common light. These colors, which are of different orders as regards tint, constitute what is called Newton's scale: and bv reflected light, occur as follows, beginning with the cen- tral spot. First order; 1st order. Second; 2d order. Third; 3d order. Fourth, &c.; 4:th order. 5th order. 6th order. Seventh. 7th order. Black, very faint blue, brilliant white, yel- low, orange and red. Dark violet, blue, yellow-green, bright yel- low, crimson and red. Purple, blue, rich green, fine yellow, pink and crimson. Dull blue-green, pale yellow-pink, and red. Pale blue-green, white and pink. Pale blue-green, pale pink. The same as 6th, very faint. The other or- ders being too faint to be distinguishable. Waves which interfere to produce the colors by reflexion: Fig. 74. Illustration and explanation; JU±----- These colors arise from the in- terference of waves reflected from the first, with those reflected from the second surface of the air plate. Suppose a small beam incident perpendicularly or nearly so, on the first surface MX of the plate, where the thickness is t. A part A 0 will be reflected back, the rest A B, being transmitted, will traverse the thickness t. At the second surface, again a part B C, is reflected, and the reflected portion return- ing through the thickness t, will emerge at the first sur- face in the direction C 0, and be superposed on that first reflected at this surface, and these will either conspire and reinforce each other or will interfere and partially or wholly neutralize each other, according to any of the conditions explained in § 7, depending upon the differ- ELEMENTS OF OPTICS. 287 ence of route 2 t. Whenever 2 t is equal to any even if 21be an even multiple of \ X, for any color, this color will be increased, multlPleof **» and when equal to any odd multiple of \ \ it will be suppressed. Now, 2 t, will vary from a value sensibly nothing to one equal to many times X, for even the long- est waves, in passing outward from the point in which th-e spherical surfaces are tangent to each other, and if an odd hence the colored fringes and the intermediate darkmultiple* rings. But the portion reflected at the second surface will, Transmitted in part, be again reflected at the first, and will traverse ^en reflected the thickness t, a third time, and emerge below super- rins8 are dark; posed upon the portion first transmitted at the second surface. The difference of route of these portions will also be 2 t, so that the effects should be the same on either side of the lenses. Experiment shows, however, that this is not the case, for wherever there is total darkness by reflexion, there is a maximum of bright- ness by transmission. Hence, there must be half an interval subtracted from the route at each internal re-*1 flexion ; the cause of the loss being a change in density and elasticity at the surfaces of contact of the glass and air. This will give for the interfering rays, in case of reflected rings, a difference of route expressed by accounted for: T 9, ' Difference of route for reflected rings; and for the transmitted, 2 t + X. To ascertain the value of t, at the different rings, call d, the diameter 2PII, of one of them, as determined by actual measurement; r and r' the radii of the surfaces, v and v', the cor- responding versed sines of the arcs whose sines P H and P' H, are equal to the semi-diameter of the ring in question. Same for transmitted rings. To find t, at the different rings; 288 NATURAL PHILOSOPHY. Then, for very small arcs, we have Equation from the figure ; V = -x—) 2 r and Another equation; Value of t; whence 4 ' d* I1 L\ Same for first bright ring; In this way Newton found the thickness at the brightest part of the first ring nearest the central black spot, to be 0,00000561 of an inch. He also found the diameters of rfliame^of011 the darkest rings to be as the square roots of the even dark and bright numbers 0', 2,4,6, &c., and those of the brightest as the square roots of the odd numbers 1,3,5,7, &c. The radii of the surfaces being great compared with the diameters of the rings, the value of t at the alternate points of greatest obscurity and illumination are as the natural numbers Law of variation of t, at the dark and bright rings; liule. Above results compared with X for yellow; 0,1,2,3,4, &c, hence, the value of t, just found, multiplied by these num- bers, will give the thickness at the different rings. On comparing the value for the thickness at the first bright ring, with the numbers in the table of article (95), it will be found just equal to one-fourth of the interval denoted by X? for the yellow ray, which is the most illuminating of the elements of white light. Taking this value for t, we shall have for the difference ELEMENTS OF OPTICS. 289 of route for the interfering rays producing the dark Difference of rings by reflexion, including the central black spot, route for the dark r< rings; X 3 x 5 x 7 x „ 2~' ~2~'~¥' "2"' TC'5 these being the even multiples of \ X, increased by \ X for the retardation caused by one internal reflexion. The odd multiples, increased by \ X, give X, 2 X, 3 X, &C. Same for the bright reflected rings. The transmitted rings will be complementary to those seen by reflexion. The phenomena we have just considered are equally Same produced, whatever may be the medium interposedphenomena between the glasses, the only difference being in the different media. contraction or expansion of the rings, depending upon the refractive index of the medium. It is found that as the refractive index of the medium increases, the diame- ter of the rings will decrease, which might have been inferred from article (99). § 105. If any one of the rings at a particular color be conceived to be expanded in all directions in the plane of the ring and to retain the same thickness, it is obvious that the plate thus produced would present the same color over its entire surface. If a second plate of the colors of natural ,t»T i . • i i ■% i i i • j ,i • ., bodies explained. same thickness and material be placed behind this one, it would act upon the waves transmitted through the first just as the latter did upon the incident waves, and the same would be true of any number of plates, so that a body made up of a series of such plates would present a uniform, distinct, and characteristic color. These con- siderations, in connection with those relating to the color of minute gratings or striae, furnish an explana- tion of the colors of natural bodies. 19 290 NATURAL PHILOSOPHY. COLORS OF INCLINED GLASS PLATES. Colors of inclined glass plates; Circumstances attending the deviation of light by such plates; Emergent waves will generally have travelled routes differing in length ; Two will emerge after having travelled different routes of equal length; Illustration; § 106. If a luminous object be viewed through two plates of glass of precisely equal thickness, slightly in- clined to each other, it will be evident that besides the transmitted image, there will be a number of images formed by the successive reflexions between the glasses. The first or brightest of these is formed by the waves whicli have all undergone two reflexions and at different pairs of the four surfaces. On entering the first plate they undergo a partial reflexion at every surface they successively encounter, each of the reflected waves un- dergoing a similar series of partial reflections at each surface. Thus it will appear evident that the different portions into whicli the waves have been separated must go through a length of route differing by the length of the interval between the glasses and the thickness of the glasses, or the different multiples of those which they have respectively traversed. They will, therefore, in genered, emerge after traversing routes whicli differ by considerable quantities. Among these portions, however, there are two which, (if we abstract the very small difference in the in- terval between the glasses at the points where they re- spectively pass,) will have gone through different routes of precisely equal length. These two waves will be, 1st. One which passes di- rectly through the first plate A B, equal to t, and through the interval B C, equal to /, between the plates, is then reflected at C, in the first surface of the second plate, re- turns along CD, equal to i, and a thickness D E, equal to the first, or t; Fig. 76. ELEMENTS OF OPTICS. 291 at the first surface it is reflected again and passes the Nvhole system EF+ F G + GII, equal to 2 t + t; 0r Explanation; upon the whole, it has travelled over 4 t + 2 i. 2d. Another portion proceeds directly through the whole A B -f B C + Cd, equal to 2 t + i, is reflected at d, in the last surface, retraces the distance de + ef, equal to t + i, is reflected at the second surface of the first glass and pursuing the course f g + g h, equal to i + t, emerges after having, on the whole, passed through i t + 2 i, or a route exactly equal in length to that of the former, neglecting, as before, the difference in i. It will be seen that out of all the possible combina- No other waves tions of different successive reflexions, these two are the™ndi"ion. ' only ones which will give routes precisely equal; all the others will differ by quantities amounting to some mul- tiple of t or i. If we now recur to the small difference in the interval i, for the points at whicli the rays respec- tively pass, it is obvious that by slightly altering the in- clination of the plates we may diminish the difference of ex^ib^ti^ routes to any amount, and may consequently make them colored fringes. differ by half a wave length, or any multiple of the same; and we shall thus produce colored fringes sepa- rated by dark bands, parallel to the intersection of the planes of the glasses. COLORS OF THICK PLATES. § 107. Another phenomenon, which depends upon the same principle, and called the colors of thick plates, will be readily understood from pre- ceding: considerations. The effect is observed to take place under these circumstances, viz. : Light being transmitted through a small hole A, in a screen, and al- lowed to fall npon a spherical con- Colors of thick plates; How they may be exhibited; 292 NATURAL PHILOSOPHY. Facts with regard to these colors; How produced. niustration and cave glass reflector M X, with con- *>• 77- explanation; centric surfaces, the back being sil- vered, and its centre of curvature situated at the aperture, there will be formed upon the screen about the aperture a series of colored rings, or in luminous waves of a single length, alternate bright and dark cir- cles. These become faint and disap- pear if the distance of the screen be increased or diminished beyond a small difference from its original position. They diminish in diameter as the glass is thicker. They arise from the interference of the waves reflected from the back surface with those reflected from the front. Denote by y, the radius A D, of one of the rings, either dark or bright; by t, the thickness CE, of the reflector; and by r, the radius A C. The equivalent interval to t, in air will be m t, in which m denotes the relative index of refraction of air and glass. Then Notation; Equation; CD= ^T2 + y2 = r + y 2r neglecting the 4th and higher powers of y. In like manner, Another ED = y2 V(mt + ry + y* =mt + r + 2 , t + ry Difference of routes equal to some multiple of and since the difference of the routes A C + CE+ ED, and A C + C D, must equal some multiple of one-half the wave length, we have, because A C is the same in both routes, mt+mt+r+ y2 2(mt + r) 2r = n. \ X, s ELEMENTS OF OPTICS. 293 whence __ /(4mt — nX) .r .(mt + r) Yalueforradiu3 y — \/---------------:-----------) of the assumed ™>t ring; and neglecting ?nt, being insignificant as compared with r, __ /(4 mt--n X) Same reduced. y — ^ >/-------7----.... (yi) m t This accords precisely with the most exact measure- ments of Sir Isaac Newton. , r sin ■ • X n .* r . n • y y) X Eelations from —. = —t- = -^— the figure; <*> fr Jv whence we deduce + x=fr-fv=l. (/r +/„) Same reduced; w a 304 NATURAL PHILOSOPHY. Radius of circle ^* of least chromatic aberration; y fr +A (97) The denominator of this expression is equal to twice the mean value of f", and therefore, Diameter of the same; 2y = «(/r_/,). I-,; Measure of chromatic aberration. and from Equation (27), we have mv — 1 mr — 1 mv — mr Jv Jr or /* y? ^v ^r /*"2 r — J v —-------------• J 3 by substituting f"2, for fr.fv, to whicli it is nearly equal. Substituting the value of p, from second equation of group (30), the above becomes Same in a different form; Jr Jv — m„ — m. / "a m-1 F. II Final value for diameter of circle of least chromatic aberration. hence, 2y = a.m*~mr f If m-1 F. = a.D. '- f" II F. II (98) In the case of parallel rays, the last factor is unity, from which we conclude, that the diameter of the circle of least chromatic aberration is equal to the radius of aperture of the lens, multiplied by the dispersive power. The distance of this circle from the lens is, ELEMENTS OF OPTICS. 305 Distance of this /» .uisiance 01 mis v * y circle from the a tens; replacing JL by its value in Equation (97), we have f 4.x- J^fvfr /QQ\TheSame Jv~r*— --n , /......\?y) reduced. Jr +Jv The effect of chromatic aberration is to give color toEffectof the image of an object, and to produce confusion ofcnromatic vision in consequence of the different degrees of conver-aberratlon; gence in the differently colored waves proceeding from the same point of an object. The vertices of the cones composed of the rays of these waves, lying in the axis, every section perpendicular to this line will have its brightest point in the centre, and the yellow waves con- in part verging nearly to the mean focus, and having by far the destroycd; greatest illuminating property, the bad effects which would otherwise arise from this aberration are in part destroyed. Besides, these effects may be lessened by reducing the aperture of the lens, though not in the Ma be same degree as those arising from spherical aberration, diminished. ACHROMATISM. § 119. It is, then, impossible, by the use of a single Achromatism; homogeneous lens, to deviate the different waves of white light accurately to a single focus, and, consequently, im- possible, by the use of such a lens, to form a colorless image of any object; both, however, may be done by the union of two or more lenses of different dispersive powers. The principle according to which this maybe accomplished, Achromatic is termed Achromatism, and the combination is said toc be achromatic. Let us suppose two lenses of different dispersive powers placed close together. The focus of the combination will, 20 306 NATURAL PHILOSOPHY. Two lensea Equation (34), and the fourth Equation of group (30), for taken; any one 0£ fa^ elementary colors as red, be given by Focus for red; 1 mr—l mrf — 1 1 fr P t J and for violet, Focus for violet; /. // mv - 1 mv' — 1 , 1 --------1------?----r P P / ' Equating these focal distances; If fr" and fv", were equal, the chromatic aberration, as regards these colors, would be destroyed; equating them we have, (mr - 1) P'+ « - 1) P = (mv - 1) P' + « - 1) P whence, Relation obtained; _P_ (mv -1)- {™r ~ 1) _, _ mv ~ m, P' (m'r-l)-K'-l) m„ — m„ J- 5 Same in a different form; the second member being negative, because m'v, is greater than m'r. i___-j Multiplying both members of this equation by------ m — 1' it may be put under the form, mf — i i P m — i mv — mr m • 1 m'v m'r . . (100) m—1 Explanation of The second member expresses the ratio of the disper- the result; giye powers of the media, and the first, the inverse ratio of the powers of the lenses for the mean waves; this being negative, one of the lenses must be concave, the other convex; and the powers of the lenses being inversely ELEMENTS OF OPTICS. 307 as the focal distances, we conclude, that chromatic aber-Rule for constructing: an ration, as regards red and violet, may be destroyed by a uniting a concave with a convex lens, the principal focal for red and lengths being taken in the ratio of their dispersive powers.^ The usual practice is to unite a convex lens of crown glass with a concave lens of flint glass, the focal distance usual of the first being to that of the second as 33 to 50,com ina 10n; these numbers expressing the relative dispersive pow- ers as determined by experiment; (see Table § 116). The convex lens should have the greater power, and, there- fore, be constructed of the crown-glass; otherwise, the effect of the combination would be the same as that of convex lens a concave lens with whicli it is impossible to form ashould have the x greater power; real image of a real object. Fig. 84. -^------- mr ■p* @Z"''^jfc' Illustration; 4&- To illustrate : let parallel rays be received by the lens A ; its action alone would be, to spread the different colors over the space VE, whose central point m, is dis- tant from A, 33 units of measure, (say inches), the violet being at V, and the red at R\ the action of the lens B, alone would be, to disperse the rays as though they pro- ceeded from different points of the line V R\ whose Explanation of central point m!, is distant from B, 50 inches, the violet ^^ ^ appearing to proceed from V, and the red from R'; and the effect of their united action would be, to concentrate the red and violet at F, whose distance from the lens is equal to the value of F, deduced from the formula 1_ F 1_ 33 + 50 97 , 06 inches. Example; 308 NATURAL PHILOSOPHY. Point in which q\* red and violet would be united; F = — 97 , 06 inches. Geometrical illustration; Fig. 84. V J£^ A' 'Why the other colors would not generally be concentrated in the 6ame point; Secondary spectrum. Now, any one of the colors, orange for example, at 0, in the space R V, which is thrown by the convex lens in advance of the centre m, and the same color at 0' in the space V R', which is thrown by the concave lens behind the centre ml, will, it is obvious, be united with the violet and red at F, by the joint action of both lenses ; and the same would be true of any other color, but for the irrationality of dispersion of the me- dia of which these lenses are composed, which prevents it, and hence an image formed by such a combination of lenses will be fringed with color; the colors of the fringe constituting what is called a secondary spectrum. An additional lens is sometimes introduced to complete the achromaticity of this arrangement. S'lW.UIK't'S which fulfil the conditions for perfect achromaticity; § 120. If two lenses, constructed of media between whicli there is no irrationality of dispersion, be united according to the conditions of Equation (100), the com- bination will be perfectly achromatic. It is found that between a certain mixture of muriate of antimony with muriatic acid, and crown-glass, and between crown-glass and mercury in a solution of sal ammoniac, there is lit- tle or no irrationality of dispersion. These substances have therefore been used in the construction of com- pound lenses which are perfectly achromatic. The figure ELEMENTS OF OPTICS. 309 represents a section of one of these, Fis-85- consisting of two double convex lenses of crown-glass, holding be- tween them, by means of a glass cylinder, a solution of the muriate in the shape of a double concave lens, the whole combined agreeably to the relations expressed by Equation (100). The focal distance of the convex lenses is determined from Equation (31). Representation of an achromatic lens; §121. From Equation (98), we infer, that the circle circle of least of least chromatic aberration is independent of the focal ^™^ length of the lens, and will be constant, provided the independent of aperture be not changed. Now, by increasing the focalfocal length; length of the object glass of any telescope, the eye lens remaining the same, the image is magnified; it follows, therefore, that by increasing the focal length of the field lens, we may obtain an image so much enlarged that the color will almost disappear in comparison. Besides, Telescopes an increase of focal length is attended with a diminution formerly ™r long; of the spherical aberration. This explains why, when single lenses only were used as field lenses, they were of such enormous focal length, some of them being as much as a hundred to a hundred and fifty feet. The use of achromatic combinations has rendered such lengths unnecessary and reduced to convenient limits, instru- Modem ones ments of much greater power than any formerly made s with single lenses. INTERNAL REFLEXION. § 122. Whenever the waves of light in their motion ^^ through any medium meet with a change of density and reflexion; elasticity, they will be both reflected and refracted. In consequence, objects seen by reflexion from a plate of 310 NATURAL PHILOSOPHY. When objects seen by reflexion from glass appear double; Relative brightness of the images when the second surface is in contact with various substances; glass, in the atmosphere, appear double when the faces of the glass are not parallel, there being an image formed by reflexion from each face. The image from the second surface will be brighter in proportioii as the obliquity or angle of incidence of the incident waves becomes greater. If the second surface of the glass be placed in contact with water, the brightness of the image from that surface will be diminished ; if olive oil be substituted for the water, the diminution will be greater, and if the oil be replaced by pitch, softened by heat to produce accurate contact, the image will disappear. If, now, the contact be made with oil of cassia, tbe image will be restored ; if with sulphur, the image will be brighter than with oil of cassia, and if with mercury or an amalgam, as in the common looking-glass, still brighter, much more so indeed than the image from the first surface. The mean refractive indices of these substances are as follows : Refractive indices of these substances; Air,.....1,0002 Water, - - - - 1,336 Olive Oil, - - - 1,470 Pitch, - - 1,531 to 1,586 Plate Glass, 1,514 to 1,583 Oil of Cassia, - - 1,611 Sulphur, - - - 2,118 indices Taking the differences between the index of refraction for compared with p]ate giass an(i ti10Se f01. t}ie other substances of the the index for plate glass; table, and comparing these differences with the forego- ing statement, we are made acquainted with the fact, which is found to be general, viz. : that when two media are in perfect contact, the intensity of the light reflect- ed at their common surface will be less, the nearer their refractive indices approach to equality; and when these are exactly equal, reflexion will cease altogether. This Conclusion. . & is an obvious consequence of the rationale of reflexion, given in Acoustics, § 71. ELEMENTS OF OPTICS. 311 surface. §123. Different substances, we have seen, have, in owing to a i -i. rv> -i. • m t difference of general, different dispersive powers, lwo media may, dispersive therefore, be placed in contact, for each of which the power the light _ _ will not all be same color, as red, for example, may have the same in- transmitted at dex of refraction, while for the other elements of white the second light, the indices may be different; when this is the case, according to what has just been said, the red would be wholly transmitted, while portions of the other colors would be reflected and impart to the image from the second surface the hue of the reflected beam; and this would always occur, unless the media in contact pos- sessed the same refractive and dispersive powers. ABSORPTION OF LIGHT. §124. The waves of light which enter any body are Absorption of not transmitted without diminution; but in consequence ofllgh a want of perfect elasticity due to the reciprocal action of the molecules of the ether and the particles of the body, and owing to the absence of perfect contact of the elements of bodies, these waves undergo a series of internal reflexions which give rise, as in the case of how produced. sound, to interferences and consequent loss of intensity. This action of bodies upon light is called absorption. The quantity absorbed is found to vary not only from Quantity one medium to another, but also in the same medium absorbed varies; for different colors; this will appear by viewing the pris- matic spectrum through a plate of almost any transpa- rent, colored medium, such as a piece of smalt blue glass, when the relative intensity of the colors will appear al- tered, some colors being almost wholly transmitted, while others will disappear or become very faint. Each color may, therefore, be said to have, with respect to every medium, its peculiar index of transparency as well as of ™«**ncy. refraction. 312 NATURAL PHILOSOPHY. Quantity absorbed depends upon; Extreme colors transmitted longest. Herschel's hypothesis to account for the extinction of a homogeneous wave; The quantity of each color transmitted, is found to depend, in a remarkable degree, upon the thickness of the medium; for, if the glass just referred to be extremely thin, all the colors are seen; but if the thickness be about 2V of an inch, the spectrum will appear in detached portions, separated by broad and perfectly black inter- vals, the rays corresponding to these intervals being to- tally absorbed. If the thickness be diminished, the dark spaces will be partially illuminated; but if the thickness be increased, all the colors between the extreme red and violet will disappear. Sir John Hersciiel conceived that the simplest hypothesis with regard to the extinction of a wave of homogeneous light, passing through a homogeneous me- clium is, that for every equal thickness of the medium traversed, an equal aliquot part of the intensity which up to that time had escaped absorption, is extinguished. That is, if the---th part of the whole intensity, which m will be called c, of any homogeneous wave which en- ters a medium, be absorbed on passing through a thick- ness unity, there will remain, Portion transmitted through a unit of thickness; n m—n c----c = - m m u 3 n and if the---th part of this remainder be absorbed in m passing througli the next unit of thickness, there will remain Portion transmitted through two units; m — n n(m — ri) m m m — n m- and through the third unit, Through three units; m — n2 n(m — n)2 t Im — m- m (m — n\3^ =-----) c, \ m I ELEMENTS OF OPTICS. 313 and through the whole thickness denoted by t units, Im — nV"1 n (m — n\~x (m — nV Through t units 1---------j • C------(---------J • C =(--------\.C. of thickness; \ m I m\ m I \ m I So that, calling c the intensity of the extreme red waves in white light, cf that of the next degree of re- frangibility, c" that of the next, and so on, the incident light will, according to Sir J. H., be represented in in- tensity by C + C' + C" + C'" + &C. Intensity of incident light; and the intensity of the transmitted light, after travers- ing a thickness t, by cyl +c' y'1 + c" y'rt + &c. . . . (101)Th^ of transmitted light; Wherein y, represents the fraction_______, which will m depend upon the waves and the medium, and will, of course, vary from one term to another. From this it is obvious, that total extinction will be Total extinction impossible for any medium of finite thickness; but if finite0 the fraction y, be small, then a moderate thickness, whicli Sickness; enters as an exponent, will reduce the fraction to a value perfectly insensible. Numerical values of the fractions y, y', y", &c, may indices of be called the indices of transparency of the differenttransparenc^ waves for the medium in question. There is no body in nature perfectly transparent, though No bod>r in all are more or less so. Gold, one of the densest of me- ^parent? tals, may be beaten out so thin as to admit the passage of light through it: the most opaque of bodies, charcoal, becomes one of the most beautifully transparent under a different state of aggregation, as in the diamond, " and all colored bodies, however deep their hues and however seemingly opaque, must necessarily be rendered visible by waves which have entered their surface; for if re- flected at their surfaces they would all appear white 314 NATURAL PHILOSOPHY. colors of bodies alike. Were the colors of bodies strictly superficial, no not superficial; Powders and streaks. variation in their thickness could affect their hues ; but so far is this from being the case, that all colored bodies, however intense their tint, become paler by diminution of thickness. Thus, the powders of all colored bodies, or the streak they leave when rubbed on substances har- der than themselves, have much paler colors than the same bodies in mass." THE RAINBOW. Eainbowdefined; § 125. The rainbow is a circular arch, frequently seen in the heavens during a shower of rain, in a direction from the observer opposite to that of the sun. If A B C, be a section of a prism of Fl°' 86, water at right angles to its length by a ver- Illustration by ° d prisms of water; tical plane, and Sr a beam of light pro- ceeding from the sun; a part of the latter will be refracted at r, reflected at D, and again refracted at r, where the constituent elements of white light, which had been separated at r, will be made further divergent, the red taking the direction of r' R, and the violet the direction r' V, making, because of its greater refractive index, a greater angle than the red with the normal to the refracting surface at r'. To an observer whose eye is situated at E, the point r' will appear red, the other colors passing above the eye ; and if the prism be de- orderinwhich passed so as to occupy the position A'B'C, makino- the colors will ff x ^ x ? o appear in the r V , parallel to r' V, the point r" would appear of a primary bow; vi0iet j^ tjie remaining colors from this position of the ELEMENTS OF OPTICS. 315 prism falling below the eye. In passing from the first Prisms replaccd t , . r& by drops of to the second position, the prism would, therefore, pre-water; sent, in succession, all the colors of the solar spectrum. If, now, the faces of the prism be regarded as tangent planes to a spherical drop of water at the points where the two refractions and intermediate reflexion take place, the prism may be abandoned and a drop of water sub- stituted without altering the effect; and a number of these drops existing at the same time in the successive po- sitions occupied by the prism in its descent, would exhi- bit a series of colors in the order of the spectrum with the red at the top. Aline ES, passing through the eye and the sun, is Axis of always parallel to the incident rays; and if the vertical vertical pl. me; plane revolve about this line, the drops will describe con- centric circles, in crossing which, the rain in its descent will exhibit all the colors in the form of concentric arches having a common centre on the line joining the eye and the sun, produced in front of the observer. When this line passes below the horizon, which will always be the case when the sun is above it, the bow will be less than is seini-circuiar, a semi-circle ; when it is in the horizon, the bow will be &c > semi-circular. Fig. 87. Illustration for primary bow, To find the angle To find the angle subtended at the eye by the radii subtended at ^ of these colored arches, let A B D, be a section of a drop ^ ^thc of rain through its centre: SA the incident, AD the colored arches; 316 NATURAL PHILOSOPHY. Fig. 87. Illustration for primary bow; Notation; Equation for one internal relkxion ; For two internal reflexions; refracted, D B the reflected, and B R the emergent ray. Call the angle CAm— the angle of incidence, ' and this, in Equation (102), gives, on reduction, b = 2{n + l)y'-{n-x)« . . . .(103) Because the chords are all equal, the last angle of in- cidence CB D, within the drop in Fig. (87), or CBD', ELEMENTS OF OPTICS. 317 in Fig. (88), is equal to the angle of refraction CAD, Angleof emergence eoual and hence the angle of emergence CBmr, is equal to to angle of the angle of incidence CAm. incidence; The angle A 0 B, in Fig. (87), is the supplement of the total deviation of the emergent from the incident rav, and is equal to the angle BFF, subtended by the ra-References to dius of the bow; in Fig. (88), it is the excess of total de- figures; viation above 180°. Calling this angle £, we shall have Notation and equation; 5=^(29 — 0); Fig. 88. Illustration for secondary bow; the upper sign referring to Fig. (87), and the lower to Fig. (8S); replacing 0, by its value in Equation (103,) the above reduces to General value 5 = zrz(2(p-2(n + l) a maximum §127. By means of the calculus it is easily shown and ml^mum7 *^a^ ^ *n Equation (104), is a maximum for the primary forth:secondary; and a minimum for the secondary. This explains the ELEMENTS OF OPTICS. 321 remarkable fact that the space between these bows always Eemarkabl° x d appearance of appears darker than any other part of the heavens in the the heavens vicinity of the bow ; for, no light twice refracted and once *etween these A J ' ° bows accounted reflected can reach the eye till the drops arrive at the prim- for. ary, and none which is twice refracted and twice reflected, can arrive at the eye after the drops pass the secondary ; hence, while the drops are descending in the space be- tween the bows, the light twice refracted with one or two intermediate reflexions, will pass, the first above, and the second below or in front of the observer. The same discussion will, of course, apply to the lunar rainbow which is sometimes seen. §128. Luminous and colored rings, called halos, areHalos; occasionally seen about the sun and moon; the most re- markable of these are generallv at distances of about twenty-two and forty-five degrees from these luminaries, and may be accounted for upon the principle of unequal refrangibility of light. They most commonly occur in cold climates. It is known that ice crystallizes in minute prisms, having angles of 60° and sometimes 90°: these Theirapi;erarance r 7 ° ° 7 accounted for; floating in the atmosphere constitute a kind of mist, and having their axes in all possible directions, a number will always be found perpendicular to each plane pass- ing through the sun or moon, and the eye of the obser- ver. One of these planes is indicated in the Figure. Sm* being a beam of light pa- Fig. 90. rallel to S E, drawn through the $ v S0^^ < Illustration and sun and the eye, gr Jk*» „s explanation; and incident upon the face of a prism whose refracting angle is 90° or 60°, we shall have the value of S, corresponding to a minimum from Equation (12), by substituting the proper values of m for ice. Thc mean value being 1,31, we have 21 322 Example first, NATURAL PHILOSOPHY. sin i (S + 60°) = 1,31. sin 30' i S = 40° 55' 10" — 30° = 10° 55' 10' o ka'on" S -21° 50' 20 and Example second; sin \ (8 + 90°) = 1,31. sin 45° i5 = 67° 52' — 45° = 22° 52' 5 = 45° 44'. Other phenomena of a similar nature will be noticed hereafter. Retrospective view of the phenomena of un polarized li-ht. Remarks on the disturbance of molecular equilibrium; POLARIZATION OF LIGHT. § 129. "We have thus far been concerned with the pro- pagation of luminous waves through homogeneous media, with the deviation which these waves undergo on meet- ing with a change of density, and with the superposi- tion of two or more waves, by which their effects are increased, diminished, or totally destroyed. We now come to a class of optical phenomena whose explanation depends upon considerations affecting the particular mode of molecular vibrations in these waves. When an ethereal molecule is displaced from its posi- tion of equilibrium, the forces of the neighboring mole- cules are no longer balanced, and their resultant tends to drive the displaced molecule back to its position of rest. The displacement being supposed very small in comparison with the distance between the molecules, the forces thus excited will, we have seen in Acoustics, be ELEMENTS OF OPTICS. 323 proportional to the displacement; and according to prin- ciples explained in Mechanics, the trajectory described by the molecule will be an ellipse whose centre coincides with the position of equilibrium. Hence, the vibration yVdistmbe(1 x»ii iii*« i • ti molecule in ot the ethereal molecules is, in general, elliptic, and the general,describes nature of the light thence arising depends upon the re-an eUiPse J lative directions and magnitudes of the axes. These el- liptic vibrations are in planes parallel to the wave front, and consequently transverse to the direction of wave pro- pagation. The axes of the ellipses may either preserve constantly the same direction in their respective planes, or may be continually shifting. In the former case the light Distinction is said to be polarized; in the latter, it is unpolarized poiarizedand or common light. common light § 130. The relative magnitude of the axes of the ellip- Nature of the ses determines the nature of the polarization. Whendetermined. the axes are equal the ellipses become circles, and the light is said be circularly polarized^ when the lesser axis vanishes, the ellipse becomes a right line, and the light is said to be plane polarized—the vibrations being in this case confined to a single plane passing normally through the wave front. In intermediate cases the polarization is called elliptical, and its character may vary indefi- CirculaiN Plane> nitely between the two extremes of plane and circular polarization. polarization. The term polarization in optics has come to be a misno-Use of the term iner. It was introduced before the theory of luminous explained. undulations had gained much favor with the scientific world; and was intended at the time of its adoption to express certain fancied affections, analogous to the polari- ties of a magnet, conceived to exist in the material ema- nations which, according to Isewton, constituted the es- sence of light. It would be better were it replaced by some other term more expressive of the actual condition of the light; but at present this seems to be impossible, owin«* to its very general acceptation, and it is accord- ingly retained. 324 NATURAL PHILOSOPHY. illustration by a § 131. To conceive the manner in which an undulation stretched cord; maj ^Q propagated by transversal vibrations, imagine a cord stretched horizontally, one end being attached to a fixed point and the other held in the hand. If the lat- ter extremity be made to vibrate by moving the hand up and down, each particle of the cord will, in succes- sion, be thrown into a similar state of vibration, and a series of waves will be propagated along the cord with a constant velocity. The vibrations of each succeeding particle of the cord being similar to that of the first, will all be performed in the same plane, and the whole ethereal articlesw^ represent the state of the ethereal particles along a in a plane plane polarized wave. The plane of vibration is called po arize wave; ^ plane Qj? polarization. If, after a certain number of vibrations in the vertical plane, the extremity of the cord be made to vibrate in some other plane, and then in another,—and so on in rapid succession—each particle of the cord will, after a certain time, proportional to its distance from the hand, assume in succession all these varied vibrations ; and the whole cord instead of taking the form of a curve lying in one plane, will be thrown into a species of helical curve, depending on the nature of the original disturbance. condition of the gucll ig the conciition of the ethereal molecules in waves ethereal particles in common light, of common or unpolarized light. undulation When, therefore, we admit a connection among the J3^iwibJ molecules of ether, similar to that which exists among vibrations. the particles of the cord, there is no difficulty in con- ceiving how a vibration may be propagated in a direc- tion perpendicular to that in which it is executed. The particles of ether, it is true, are not held together by cohesive forces like those of a cord, but the molecular forces which subsist among them, are of the same kind, and produce similar effects. Neither the particles of the cord nor the ethereal molecules are in contact. § 132. These illustrations being understood, conceive a transversal vibration to proceed from a disturbed mole- ELEMENTS OF OPTICS. 325 Fig. 91. A Jl X' M eule at A, towards C, and suppose the vibration to take place in the plane of the pa- per, and let MNbe the front of the wave at the expiration of any time t, after the be- ginning of motion. The dis- placement x, of the molecule at C, will, § 55, Acoustics, be given by the equation w JV Transversal vibration supposed to proceed from a disturbed particle at A; X = a sin (2 tf Vt c ) Consequent displacement of another particle at C; om in which a denotes the amplitude of the disturbance at A; c, the distance from A to C\ V, the velocity of wave propagation, and X the length of the wave. At the same instant, suppose a second transversal vibra- A secoml tion to proceed from any other point, as B, towards C, vibration fr the vibrations in the latter case being perpendicular to any other point; the former, and let M' N' be the front of the wave at the expiration of the same time t, as above. The dis- placement y, of the molecule C, due to this action, will be given by the equation y =z — . sin (2 tf. c. \ Vt x ) Uisplacement of the same particle at<7; in which b denotes the amplitude of the disturbance at B, and ct, the distance from B to C. Dividing these equations respectively by the coefficient of the circular function in the second member, we obtain the equations, 2 *r . Vt— c x I . ex \ .— arc Ism =---1, Equations obtained from these displacements; 2^ . 7i?-^ = are(Bin = M.); 326 NATURAL PHILOSOPHY. the cosines of these arcs will be respectively Cosines of these _________ arcs; / C2 X2 /\ C2 y2 V 1 - —-' and V 1 - "^r- • Subtracting the second of these equations from the first, we find, Combining these 2 AT / . , . C X, , • CJ ?/ v equations and — . (<-\—C) = arc (Sin = ---)— arc (sin = L^). reducing; *• a 0 Taking the cosine of each member of the equation, and recollecting that the cosine of the difference of two arcs is equal to the sum of the rectangles of the cosines and sines, we find, after a slight reduction, We obtain the p 2 c2 ^ it, N C C • 2if / W-,~^N equation of an j-}/* + "T »"—2 C0S "T-(C ~G)' ~X' r2/=SmV V>~ €>(10 0 ellipse; b2 a2 X a b X which is an equation of an ellipse referred to its centre. Light eiiipticaiiy The axes of the ellipses in this case preserving the same poanz direction, the light will, from what we have already said, be eiiipticaiiy polarized, and is obviously compounded of two waves plane polarized in planes at right angles to each other. When Supposition in C X j C C. , — c = —, and — = Equation (107); ' 4 ' a J ' Equation (107), reduces to Which reduces it ~>2 _L y2 ___ q2 . HOS^ to the equation of ^ ' a circle; the equation of a circle, and the light becomes circnlarly polarized, being compounded of two plane polarized waves Li-htcircularly °^ e(lua^ intensity, having their planes of polarization at polarized; right angles to each other. In this latter case, the light ELEMENTS OF OPTICS. 327 will possess many of the properties of common light, but common light mav be regarded will differ from it in some important, particulars to be a3 compouude(i noticed presently. of two waves Fig. 92. polarized in planes perpendicular to each other; §133. The difference between polarized and common light being, that in the former, the axes of the ellipses de- scribed by the molecules remain parallel, while in the latter they are incessantly changing their directions; common light, like eiiipticaiiy polarized light, may be regarded as compounded of two plane polarized waves, of which the planes of polarization are at right angles to each other. When these component vibrations are separated, each component becomes plane polarized light, grating these This separation may be effected, either by causing these component component vibrations to take different directions by or- dinary reflexion and refraction, by the retardation or ac- celeration of one over the other, as in the case of double __ . 7 Different ways of refraction, soon to be explained, or by absorbing one and causing tins permitting the other to pass unobstructed. Effect of separation. POLARIZATION BY REFLEXION AND REFRACTION. . Polarization by § 131. It is ascertained that when a wave ot common reflexion and by light is incident on any transparent medium of uniform refraction; density, under a certain angle of incidence, called the polarizing angle, the resolution above referred to, takes place; the reflected and refracted waves become plane polarized, the former in the plane of reflexion, and the latter in a plane at right angles to it. Both waves lose "almost entirely the power of being again reflected or re- fracted when the surface of a second deviating medium introductory is presented to either in a particular manner. remarks; 328 NATURAL PHILOSOPHY. Fig. 93. Experimental illustration: Explanation of apparatus; Appearance when the analyzer is perpendicular to the plane of first reflexion: The same when the analyzer is revolved through any angle less than 90°; Fig. 94. Thus, MN and M! Nf, representing two plates of glass, mounted upon swing frames, attached to two tubes A and B, which move freely one within the other about a common axis, let the beam SD. of homogeneous light, be received upon the first under an angle of incidence equal to 56° ; reflexion and refraction will take place ac- cording to the ordinary law, and if the reflected beam D D', which is sup- posed to coincide with the common axis of the tubes, be incident upon the second re- flector under the same angle of incidence, the reflector being per- pendicular to the plane of first reflex- ion, it will be totally reflected, there beinsr none refracted. But if the tube B, be turned about its axis, the tube A being at rest, the angle of incidence on the glass M' N', will remain imchanged, refraction Fisr. 95. ELEMENTS OF OPTICS. 329 Fis. 96. will begin, and the re- fracted portion will in- crease while the reflected portion will diminish, till the tube B has been turn- ed through an angle equal to 90°, as indicated by the graduated circle C, on the tube A ; in which position of the reflector, the beam will be totally refracted. Continuing to turn the tube B, the reflexion from 31' N' will increase, and the refraction will decrease, till the angle is equal to 180°, when the plane of the first reflexion will be again perpendicular to M' JV\ and the whole beam will be reflected; beyond this, reflexion will again diminish, and refraction increase, till the angle becomes 270°, when the beam will be to- tally refracted; after passing this point, the same phe- nomena will recur, and in the same order, as in the second quadrant, till the tube is revolved through 360°, when the restoration of the reflected wave will be com- plete. The same phenomena would have occurred had the second reflector been presented to the refracted com- ponent of the original incident wave on its emergence from the first plate of glass. It is important to remark in this connection, that the molecalar vibrations in the wave reflected from, and in that transmitted through the second reflector, take place, the former inthe plane of second reflexion and the latter in a plane at right angles to it; and that the effect of the second reflector is, therefore, to twist, as it were, the planes of polarization of these component waves in opposite directions, that of the reflected wave through an angle which measures the rotation of the second reflector about the axis of the tubes, and that of the refracted wave through an angle which is its complement. It thus appears that a beam of homogeneous light re- flected from, or refracted through, a plate of glass, be- ino- incident under an angle equal to 56°, immediately The same for a revolution through 90°; Appearances when the analyzer is revolved through the other three quadrants. Same phenomena exhibited by the refracted wave. Important remark. 330- NATURAL PHILOSOPHY. characteristics acquires opposite properties, with respect to reflexion and polarized li-ht, refraction, on sides distant from each other equal to 90°, measuring around the beam ; and the same properties at distances of 180° ; and these among other properties dis- tingush plane polarized light. Effects when the We have supposed the angle of incidence 56°, if it — f0 ,*•«• were less or greater than this, similar effects would be ob- lncidence diners o ~ from that of served, though less in degree; or, in other words, the polarization. waves first deviated would be eiiipticaiiy polarized, the eccentricity of the elliptical orbit increasing as the angle approaches more and more to that of polarization. Fig. 93. Apparatus. Position of the plane of polarization determined; Analyzer revolved; Intensity of reflected beam; The plate MN is called the poleirizer, and IF JV, the analyzer. The position of the plane of polarization in any plane polarized wave, is readily ascertained by the total reflexion whicli takes place from the analyzer, when, the polarized beam being incident under the polarizing angle, the plane of the analyzer is perpendicular to it. Starting from this position of the analyzer, with respect to the plane of polarization, and calling a, the angle be- tween the plane of polarization and that of second inci- dence, which is equal to the angle through which the analyzer has at any time been turned about the first reflected or polarized beam ; A, the intensity of this beam, and I, the variable intensity of that reflected from the analyzer in its various positions, the formula 1= A cos a, (109) ELEMENTS OF OPTICS. 331 will express, for uncrystallized media, the law according Law expressed to which a polarized beam will be reflected from thebvformu1a; analyzer when the angle of incidence is equal to that of polarization. According to this law, if we conceive a wave of com-Thislluvai,i>liod i ° • -i -i *°commcm mon light as it emanates from any self-luminous body, light. to be compounded of two waves polarized in planes at right angles to each other, that is, supposing the orbital motion of the molecules to arise, as they will, from two component rectilinear motions at right angles to each other, Equation (107), we should have for the intensity of reflexion from a reflector, I + F = A . COS2 a + A . COS2 (90°--a ) = A, Consequence; in which I and /', denote the intensity of reflexion of the two component polarized waves; whence, the inten- sity of the reflected wave will be the same on whatever conclusion; side of the incident beam the analyzer be presented. §135. What has been said of the effects of glass rolal.izingangle on light is equally true of other transparent homogene- ^sJith the ous media, except that the polarizing angle, which is con- stant for the same substance, differs for different bodies. It is found, from very numerous observations, that the tangent of the maximum polarizing angle is edways equal to the refractive index of the reflecting medium taken in r^ reference to that in which the wave is reflected; thus, calling the relative index m, and the polarizing angle ' ? refraction will emergence. appear. § 144. One of the component waves in Iceland sparForm of the ex -»- component is propagated equally in all directions, and is, there- waves in Iceland fore, spherical in form when proceeding from a pointspar* in the crystal; the other is propagated unequally in different directions, the form of the wave being that of an oblate spheroid of revolution, whose shorter axis co- incides with the optical axis of the crystal; resembling the case taken for illustration in § 42 of Acoustics. Now, the radius of the ellipsoidal wave is always Nation between , . , 1 the radii of the greater than that of the spherical wave, except when ordinary an(1 the refracted ray coincides with the axis; and these radii extraordinary . , , -, ,t waves; being described in the same time, may be taken as the measures of the velocities of wave propagation in the extraordinary and ordinary waves. The refractive in- dex being equal to the ratio of the velocity before incidence, to Fis- m that within the crystal, the extra- ordinary index will be variable, and less than the ordinary index. X\J_S^^ Extraordinary But the index of refraction being X index variable; 340 NATURAL PHILOSOPHY. When the extraordinary index will be a minimum. also equal to the ratio of the sine of the angle of inci- dence to that of refraction, the extraordinary ray must always be thrown further from the axis than the ordi- nary ray; and the extraordinary index of refraction will have its minimum value when the extraordinary ray is perpendicular to the axis. Properties the reverse of those of Iceland spar; Fig. 101. Eockcrystal; § 145. With rock crystal, which occurs in the form of hexagonal prisms, terminated with six-sided pyramids, the case is just reversed; the ellipsoidal wave is prolate, its longer axis coinciding with the op- tical axis of the prism, and being equal in length to the radius of the spherical wave ; the extraordinary ray is always found between the ordinary ray and the axis, as if drawn towards the latter; and the extraordinary index has its maximum value when the ordinary ray is perpendicu- lar to the axis. These circumstances have given rise to a division of doubly refracting substances into two classes Doubly refracting distinguished by their axes, which are said to be posi- tive when the extraordinary ray is between the ordina- ry ray and the axis, as in the case of rock crystal; and negative, when the positions of these rays are reversed with respect to the axis, as in Iceland spar. substances classified. To find any value of the extraordinary index between the maximum and minimum. § 146. Having ascertained by experiment the value of the ordinary index, which will be represented by m0, and the maximum or minimum value of the extraor- dinary index, according as the crystal has a positive or negative axis, whicli will be represented by me; then, de- noting the space described by the wave before incidence, in the same time that the radius vector of the extraor- dinary wave is described, by unity, will the extraordi- nary index between m0, and me, be found by the fol- lowing law. Let an ellipsoid of revolution, Fig. (102), be conceived, ELEMENTS OF OPTICS. 341 having its centre C, at the point, Flg*102, ^le ; of incidence, its axis of revolution coincident with the optical axis C X, of the crystal, and its polar to its equatorial radius in the in- verse ratio of the ordinary, to the minimum or maximum value of the extraordinary index of refraction, according as th crystal belongs to the negative or positive class. Then, in all positions of the extraordinary ray, its index is eaual to the reciprocal of its length contained between Valuc °f x x o extraordinary the centre and surface of the ellipsoid. index. The equation of a section of this surface through the axis, referred to the centre, is A2y2 +B2 x2 = A2 B2, and calling r, the length of the extraordinary ray be- tween the centre and the surface, and d, its angle of inclination with the optical axis, it reduces to Equation of a section of the surface; AB AB r — ^J»8in*'4 + -^cos2& ^B2 + {A2-B2)sm2& ; Value of radius vector; denoting by me, the value of the extraordinary index sought, we have m. 1 rz ~, -• 1 \ . Value of =-- = */ Z—J- (_________)sin2d . .(112) extraordinary V V A2 + \B2 A2 I ^ J index; in which A = m. B = 1_ mt Notation; 342 NATURAL PHILOSOPHY. Nature of crystal It is obvious that the coefficient of sin2 d, is positive or determined. . ,. . negative according as the axis is positive or negative ; hence, the coefficient of sin2 0, determines the nature of the crystal. To find experimentally the maximum and minimum values of the extraordinary index. § 147. To determine the value of m0 and me, in any particular instance, it is in the first place known that the index of the extraordinary ray will be constant and equal to its maximum or minimum value, according to the nature of the bodjr, when refracted in a plane at right angles to the optical axis ; it is only necessary, therefore, to convert the crystal, by grinding, into a prism whose refracting faces shall be j^arallel to the axis, when both the ordinary and extraordinary index may be ascertained by the method explained in § (25). To distinguish between the rays, it will, in general, be sufficient to move the prism so as to give the plane of incidence a slight inclination to its length, as in that case the extraordinary ray will be thrown out of this plane, and thus become known. In Iceland spar, These values for Iceland spar; m0 = 1,6543, me = 1,4833; hence, Its nature deduced from formula 012); A = 0,60449, B = 0,67417; the ellipsoid is, therefore, oblate ; and the coefficient of sin2 0, negative. Tourmaline, beryl, emerald, apatite, &c, also belong to this class. Quartz, ice, zercon, oxide of tin, &c., give the coefficient of sin2 d positive ; they are, other substances therefore, of the positive class, and the ellipsoid is pro- classified. late ELEMENTS OF OPTICS. 343 § 148. If a plane wave Fis-103- Tr W, of common light be incident on the upper sur- face of a crystal of Iceland spar to whicli it is parallel, this wave will be resolved into two components, one of which will take the direction of and be normal to an ob- lique line P e, and will be refracted according to the extraordinary law; the other will preserve its original course and pass through without deviation. These waves will both leave the crystal normal to that plane of prin- cipal section whicli is perpendicular to its upper face, the waves themselves becoming parallel; each will be plane polarized, the plane of polarization of the ordinary wave coinciding with the plane of principal section just named, and that of the extraordinary wave being at right angles to it. If these component waves be received upon the upper ?urface of a second crystal of the same kind, and whose optical axis is parallel to that of the first, they will take the directions e' e" and o' o", parallel, respectively, to the directions P e, and P o, and will not be again divided, the first undergoing extraordinary, and the latter ordi- nary refraction ; and if the crystals be of equal thick- ness, the distance e" o", will be double e o. If either or both of the component waves whose directions are e e', and o o', had been polarized by reflexion, refraction or absorption, the action of the second prism would have been the same ; this is, therefore, another characteristic property of plane polarized light, viz.: that it will not un- dergo double refraction when its plane of polarization is either parallel or perpenelicular to the plane of principal sec- tion ; being in the former case wholly refracted according to the ordinary, and the latter according to the extraordi- Plane wave of common light incident upon a crystal of Iceland spar; Effect of this crystal. Emergent waves received upon a second crystal of Iceland spar; Another characteristic of plane polarized light 344 NATURAL. PHILOSOPHY. Reverse trne for nary law. The reverse would have been the case if the positive crystals, crvstai5 like quartz, had possessed a positive axis. The second crystal supposed to turn on its base; Effect on the ordinaay wave; Effect on the extraordinary wave; § 149. When the crystal M' JV', is turned around on its base so that the prin- cipal sections of the crys- tals, which are normal to the upper surfaces, make an angle with each other, each of the component waves of which the direc- tions are oo' and ee\ will be again divided into an ordinary and extraordi- nary wave, whose relative intensities will depend up- on the inclination of the principal sections to each other. To avoid complica- tion, let us suppose the wave moving along P e, to be arrested by sticking a piece of wafer to the lower surface of the first crystal at e\ then will the intensi- ties of the portions into which the wave moving along oo', is divided by the second crystal, be ex- pressed by the formulas 0= J..cos2 a Oe = A. sin2 a Wherein A represents the intensity of the wave o o'; Fig. 105. ELEMENTS OF OPTICS. 345 a, the angle made by the principal sections of the crys-Notation; tals; 00, the intensity of the ordinarily refracted wave; and Oe, that of the wave refracted according to the ex- traordinary law. Removing the wafer from e, and calling Ee and EQ the intensities of the extraordinary and ordinary waves into which the wave moving on Pe is separated by the second crystal, and B its intensity on leaving the first crystal, we shall, in like manner, have A> "~ B . COS a 1 Components of • • • • • \-*--*-^7 the extraordinary Ea — B . Sin2 a ) wave; Taking the sum of the four emergent waves, there will result, Sum of the four 0. + Oe + Ee + E0 = A + B. 9 c * * emergent waves. The waves 00 and Oe, in* Equations (113), are always found to be polarized, the former in the plane of princi- pal section of the second crystal, the latter in a plane at right angles to it; and the same remark being applica- ble to Eo and Ee, in Equations (114), it follows that the , -r-r .-n -• -n i i. Positions of the planes of polarization of 00 and E0 will be parallel toplanesof eaoh other, as also those of Oe and Ee. polarization. CIRCULAR POLARIZATION. § 150. We have seen, § 132, in what circular polari- zation consists. The best method of producing this, is by means of what is called FresneVs rhomb, which consists ^*Tat[on of a parallelopipedon of St. Gobain's glass cut so that produced; 346 NATURAL PHILOSOPHY. Fresnel's rhomb; Fig. IOC. Light circularly polarized. Effect of transmitting the emergent wave through a second rhomb. Circularly polarized light distinguished from common, and from plane polarized light. the angles If and IP shall each be equal to 54i°. When a plane polarized wave is incident upon the face IfJV, parallel to its front, this wave will be totally reflected at B' and R", and will emerge parallel to the face IP N'. If the plane of polarization be inclined at an angle of 45° to the plane of reflexion, the component waves will have the same intensity, their difference of phase after two re- flexions will be found equal to one-fourth of a wave length, and the emergent light will, therefore, Equations (107) and (108), be circularly polarized. If the emergent wave be now made to undergo two more total reflexions, in the same plane and at the same angle, by transmitting it through a second rhomb, placed parallel to the first, it will emerge plane poleirized ; and its plane of polarization will be inclined 45° on the other side of the plane of reflexion. This property enables us to distinguish at once a cir- cularly polarized wave from a wave of common light; the former becomes plane polarized by two total reflexions, at an angle of 54|°. On the other hand, it is at once distinguished from plane polarized light, by the circum- stance that it is divided into two wTaves of equal inten- sity by a doubly refracting crystal, whatever be the posi- tion of the plane of principal section. Effect of metallic g \$\m "\\re might naturally conjecture that the effects reflectors on plane polarized produced by metals upon the reflected light would be llght» analogous to the phenomena of total reflexion by glass and other transparent substances,—there being no tran- smitted wave in either case. It is accordingly found that when a plane polarized wave is incident upon a metallic reflector, the reflected light is eiiipticaiiy poleirized; the laws of the phenomena are, however, dif- ferent from those of total reflexion from transparent media. ELEMENTS OF OPTICS. 347 §152. It has been observed, that when a polarized General fact; wave is passed througli a doubly refracting crvstal in the direction of its optical axis, it undergoes no change, the analyzer producing the same effect after as before transmission on being offered to the same side of the rav. To this, however, there are some remarkable excep-„ x Exception to tions, one of the most conspicuous of which is exhibited tins in the case of by rock crystal, or quartz. When the analyzer is in thequartz; position affording no reflexion, the interposition of a crystal of this substance will restore a portion of light, and to cause it to disappear, the analyzer must be turned through a certain angle about the ray, the magnitude of this angle depending upon the thickness of the inter- posed quartz. As the thickness of the mineral is in-Planeof creased, the rotation of the analyzer must be continued, polarization The plane of polarization is thus twisted into a surface |nVe^0siif- a of double curvature resembling the turns of an auger, or crystal of quartz; the surface generated by the rotation of one right line about another perpendicular to it, at the same time that it has a motion of translation along this second line. It has been found that for a given thickness, the arc Arc of rotation of rotation necessary to bring the analyzer into the posi- *!*ercn'ft* «/ o J x different colors; tion of evanescence is different for the differently colored waves, this arc being given by the formula „ & * . Formula for a ■v a ' single substance; in which r represents the arc, Jc a constant, t the thick- ness, and X the wave length for the particular color. It has also been ascertained, that for some specimens Analyzer must of quartz, it was necessary to turn the analyzer in oiie^"™ecl,n direction, while for others, in an opposite direction, and directions for that a singular connection exists between this property( © i I quartz. and the right or left handed direction in which certain small faces of the crystal lean around the summit of the variety called plagiedral quartz. Other bodies, besides quartz, possess the same property, 348 NATURAL PHILOSOPHY. other bodies but in different degrees; and if two of these bodies be Pos ,stneEame interposed, the arc of rotation is that due to the sum or property; x ? difference of their thicknesses, according as they exert their effects in the same or opposite directions. Or more generally, Formula for a combination; Notation explained ; Applies also to liquids. BT=rt + r'tf + r" t" + &c, in which Bt is the rotation due to the combination; T its entire thickness; r, r\ &c, and t, t', &c, the corres- ponding quantities answering to the several individuals of the combination ; the products entering the expression with the same or different signs, according as the diffe- rent media tend to turn the plane of polarization in the same or different directions. This formula is found to hold good not only with solid crystals, but also with liquids possessing this property, when mixed together. CHROMATICS OF POLARIZED LIGHT. introductory g 153 Having explained the general phenomena of polarization and double refraction, we pass to the consid- eration of the effects produced when polarized light is transmitted through crystalline substances. The phe- nomena displayed in such cases, are among the most splendid in optics; and when we consider that through these phenomena we are enabled almost to view the in- terior structure and molecular arrangement of natural bo- dies, the importance of the subject will be apparent. First discoveries. The first discoveries in this department of science were made by Arago, in the year 1811, and the subject has since been successfully prosecuted by some of the fir6t philosophers of Europe. § 154. We have seen that when a wave of light, po- larized by reflexion, is incident upon the analyzer under the polarizing angle, no reflexion will take place when ELEMENTS OF OPTICS. 349 the plane of incidence on the analyzer is perpendicular Effect of to that on the polarizer. Now, if between the two re-"««"? r ' polarized light Sectors we interpose a plate of any double-refracting sub- through a stance, the power of reflexion at the analyzer is suddenly ^"refracting restored, and a portion of the light is reflected, the quantity depending on the position of the interposed crystal; and by this property the double-refracting structure has been detected in a vast variety of substances, in whicli the sep- aration of the two waves was too small to be directly per- ceived. § 155. In order to analyze this phenomenon, let the crys- These effects talline plate be placed so as to receive the polarized wave anal«vzed bv x x x turning the parallel to its surface, and let it be turned round in its crystal in its own own plane. We shall then observe that there are two plane; positions of the plate in which the light totally disap- pears, and the reflected wave vanishes, just as if no crystal had been interposed. These two positions are at right angles to one another; and they are those in which the principed section of the crystal coincides with Whenno 1Mltis the plane of first reflexion, or is perpendicular to it. reflected from the When the plate is turned round, from either of theseanalyzer; positions, the light gradually increases, until the princi- amount {s a pal section is inclined at an angle of 45° to the j)lane maximum. of first reflexion, when it becomes a meiximum. § 156. In these experiments the reflected light is white, colors produced But if the interposed crystalline plate be very thin, the^rpbiu most gorgeous colors appear, which vary with every change of inclination of the plate to the polarized wave. Mica and sulphate of lime are very appropriate for the exhibition of these beautiful phenomena, because they sulphateofiime; can be readily divided by cleavage into laminae of almost any required thinness. If a thin plate of either of these substances be placed so as to receive the polarized wave parallel to its surface, and be then turned round in its Appearances own plane, the tint does not change, but varies only in ^lt^1seby intensity; the color vanishing altogether when the prin-substances; 350 NATURAL PHILOSOPHY. when the light pal section of the crystal coincides with the plane of disappears and primitive polarization, or is perpendicular to it,—and, when it is a l . L . ' .... maximum. reaching a maximum, when it is inclined to the plane of primitive polarization at an angle of 45°. The crystal fixed g 157. If, on the other hand, the crystal be fixed, and and the analyzer . tumed; the analyzer be turned, so as to vary the inclination of the plane of the second reflexion to that of the first, the color will be observed to pass, through every grade of the same tint, into the complementary color ; it being Positions eivin* always found that the light reflected in any one position complementary 0f the analyzer is complementary, both in color and in- tensity, to that which it reflects in a position 90° from the former. This curious relation will appear more evi- dently, if we substitute a double refracting prism for the analyzer; for the two waves refracted by the prism have their planes of polarization—one coinciding with the noubierefracting principal section of the prism, and the other at right an- substituted for &es ^° ^? an(l are therefore in the same condition as the the analyzer; light reflected by the analyzer, with its plane of reflex- ion successively in these two positions. In this manner the complementary colors are seen together, and may be easily compared. But the accuracy of the relation sta- ted is completely established by making these two waves partially overlap ; for, whatever be their separate tints, it Effect of causing the w^l ^e found that the part in which they are superposed tints to overlap, is absolutely white. rffoct of plates of § 158- When ]amin^ of different thicknesses are inter- variable thicknesses; posed between the polarizer and analyzer, so as to re- ceive the polarized wave parallel to their surfaces, the tints are found to vary with the thickness. The colors pro- duced by plates of the same crystal, of different thick- nesses, follow, in fact, the same law as the colors reflect- ed from thin plates of air ; the tints rising in the scale Law followed by as the thickness is diminished, until finally, when this thick- the colors; , J 1 ness is reduced below a certain limit, the colors disap- pear altogether, and the central space appears blade, as ELEMENTS OF OPTICS. 351 when the crystal is removed. The thickness producing Results of corresponding tints is, however, much greater in crystal- experiments; line plates exposed to polarized light, than in thin plates of air, or any other medium of homogeneous structure. The black of the first order appears in a plate of sul- phate of lime, when the thickness is the ^ oV o 0I> an incb 5 between „V <*• an(l i\ °f an incll5 we ^ave ^ie wllole succession of colors of Newton's scale; and when the thickness exceeds the latter limit, the transmitted light ^^ ^ is always white. The tint produced by a plate of mica, different in polarized light, is the same as that reflected from asubstance3 r ° 7 compared. plate of air of only the T£oth Part of the thickness. The same subject has been investigated for oblique in- 0blique cidences, and the laws which connect the tint developed incidences. with the number of wave lengths and parts of a length within the crystal, for a wave of given refrangibility, have been determined, both for uniaxal and biaxal crys- tals. §159. Let us now apply the principles already estab-App]ication of lished, to explain the appearances. preceding -r,, , t r» tii principles; It has been shown, that a wave ot common light, on entering a crystalline plate, is resolved into^two waves, which traverse the crystal with different velocities, and in different directions. One of these wraves, therefore, will lag behind the other, and they will be in different phases of vibration at emergence. When the plate is thin, this Preliminary retardation of one wave upon the other will amount only to a few wave lengths and parts of a length; and it would, therefore, appear that we have here all the condi- tions necessary for their interference, and the consequent production of color. But here we are met by a difficulty. So far as this An apparent explanation goes, the phenomena of interference and ofdlfficulty aris color should be produced by the crystalline plate alone, and in common light, without either polarizing or ana- lyzing plate. Such, however, is not the fact; and the real difficultv in this case is,—not so much to explain 352 NATURAL PHILOSOPHY. Its solution ; Inquiry suggested. how the phenomena are produced, as to show why they are not always produced. In seeking for a solution of this difficulty, it may be remarked, that the two waves, whose interference is sup- posed to produce the observed results, are not precisely in the condition of those whose interference we have hitherto examined; they are polarized, and in planes at right angles to each other. We are led, then, to inquire whether there is anything peculiar to the interference of polarized waves which may influence these results; and the answer to this inquiry will be found to remove the difficulty. Experimental § 160. The subject of the interference of polarized light InteXenSceofhe was examined5 ^itli reference to this question, by Fues- poiarized light; nel and Arago, and its laws experimentally developed. It was found that two waves of light, polarized in the same plane, interfere and produce fringes, under the same circumstances as two waves of common light;—that when the planes of polarization of the two wTaves are inclined to each other, the interference is diminished, and the fringes decrease in intensity; and that, finally, when the angle between these planes is a right angle, no fringes whatever are produced, and the waves no lon- ger interfere at all. These facts may be established by taking a plate of tourmaline which has been carefully worked to a uniform thickness, cutting it in two, and placing one-half in the path of each of the interfering Euies deduced, waves. It will be thus found that the intensity of the fringes depends on the relative position of the axes of the tourmalines. When these axes are parallel, and con- sequently the two waves polarized in the same plane, the fringes are best defined; they decrease in intensity when the axes of the tourmalines are inclined to one Experimental anotlier '-> and5 finally, they vanish altogether when these illustration. axes form a right angle. % 161. The non-interference of waves, polarized in ELEMENTS OF OPTICS. 353 planes at right angles to 6ne another, is a necessary result Experiments confirm the of the mechanical theory of transversal vibrations. In mechanlcal fact, it is a mathematical consequence of that theory, that theory of the intensity of the resultant light in that case is constant, Tibratious and equal to the sum of the intensities of the two compo- nent waves, whatever be the phases of vibration in which they meet. But although the intensity of the light does not vary with the phase of the component vibrations, the character of the resulting vibration will. It appears from Equation (107), that two rectilinear and rectangular vibrations com- pose a single vibration, which will be also rectilinear when the phases of the component vibrations differ by an exact number of semi-wave lengths; that, in all other cases, the resulting vibration will be elliptic; and that the ellipse will become a circle, when the component vibrations have equal amplitudes, and the difference of Jesuits of this ■*■ x i theory and their their phases is an odd multiple of a quarter of a wave experimental length. These results have been completely confirmed byconfirmatlon- experiment. In the above mentioned law we find the explanation of Apparent the fact, that no phenomena of interference or color arer^mo^J.c produced, under ordinary circumstances, by the two waves whicli emerge from a crystalline plate,—for these waves are polarized in planes at right angles to one an- other ; and we see that, to produce the phenomena of color in perfection, the planes of polarization of the two waves must be brought to coincide by the analyzer. §162. Fresnel and Arago discovered, further, thatLaw deduced ° . . , 'ii i ?rom experiment; two waves polarized in planes at right angles to each other, will not interfere, even when their planes of po- larization are made to coincide, unless they belong to a wave, the whole of whicli was originally polarized in one plane ; and that, in the interference of waves which had undergone double refraction, half a wave length must be supposed to be lost or gained, in passing from the ordi- nary to the extraordinary system,—just as in the transi- 351 NATURAL PHILOSOPHY. Another confirmation of the theory of transversal vibrations; Experiment detailed; Complementary colors. tion from the reflected to the transmitted system, in the colors formed by thin plates. The principle of the allowance of half a wave length is a beautiful and simple consequence of the theory of transversal vibrations. In fact, the vibration of the wave incident on the crystal is resolved into two within it, at right angles to one another,—one in the plane of prin- cipal section, and the other in a plane perpendicular to it. Each of these must be again resolved, in two fixed directions which are also perpendicular; and it will easily appear from the process of resolution, that, of the four components into which the original vibration is thus re- solved, the pair in one of the final directions must con- spire, while in the other, at right angles to it, they are opposed. Accordingly, if the vibrations of the one pair be regarded as coincident, those of the other must differ by half a wave length. Hence, when the plane of reflex- ion of the analyzer coincides successively with these two positions, the colors, which result from the interference of the portions in the plane of refiexion, those in the per- pendicular plane being not reflected, will be complemen- tary. Office of the polarizer; Explanation of appearances. § 163. The former of the two laws explains the office of the polarizer in the phenomena. To account mechani- cally for the non-interference of the two waves, when the light incident upon the crystal is unpolarized, we may, § 133, regard a wave of common light as composed of two waves of equal intensity, polarized in planes at right angles to one another, and whose vibrations are therefore perpendicular. Each of these vibrations, when resolved into two within the crystal, and these two again resolved in the plane of reflexion of the ana- lyzer, wrill exhibit the phenomena of interference. But the amount of retardation will differ by half a wave length in the two cases ; the tints produced will therefore be complementary, and the light resulting from their union will be white. ELEMENTS OF OPTICS. 355 § 164. The preceding laws of interference being kept Reason of the in mind, the reason of all the phenomena is apparent, phenomena; The wave is originally polarized in a single plane, by means of the polarizer; it is then resolved into two waves within the crystal, which are polarized in planes at right angles to each other; and these are finally reduced to the same plane by means of the analyzer. The two waves will, therefore, interfere, and the resulting tint will depend on the retardation of one of the waves behind the other, produced by the difference of the velocities Eesultant tmt with which they traverse the crystal. dependent upon; §165. It is plain, Equation (107), that the light issu- Emergent light, ing from the crystal is, in general, ellipAically polarized, inseneralis inasmuch as it is the resultant of two waves, in which polarized; the vibrations are at right angles, and differ in phase. Hence, when homogeneous light is used, and the emer- gent wave is analyzed with a double-refracting prism, the two waves into which it is divided vary in intensity as the prism is turned, neither, in general, ever vanish- ing. When, however, the thickness of the crystal is such thicknesses that the difference of phase of the two waves is an exacttl,e difference of -number of semi-wave lengths, they will constitute a plane ntmbeTof™0* polarized wave at emergence,—the plane of polarization semi-wave either coinciding with the plane of primitive polarization, en§t s; or making an equal angle with the principal section of the crystal on the other side, according as the difference of phase is an even or odd multiple of half a wave length. Accordingly, one of the waves into which the light is divided by the analyzing prism, will vanish in two posi- tions of its principal section; and it is manifest that the successive thicknesses of the crystalline plate, at which this takes place, form a series in arithmetical progres- sion. On the. other hand, when the difference of phase is a quarter of a wave length, or an odd multiple of that quantity,—and when, at the same time, the principal difference of section of the crystal is inclined at an angle of 45° to thepbaseis* ^ ~ quarter of a plane of primitive polarization—the emergent light will be wavelength; 356 NATURAL PHILOSOPHY. Circular polarization perfect for one color only. Color may be produced with thick plates; Method explained. circularly polarized. This is one of the simplest means of obtaining a circularly polarized wave; but it has the disadvantage, that the required interval of phase can only be exact for waves of one particular length, and that, therefore, the circular polarization is perfect only for one particular color. § 166. We have seen that the phenomena of color are only produced when the crystalline plate is thin. In thick plates, where the difference of phase of the two waves contains a great many wave lengths, the tints of different orders come to be superposed (as in the case of Newton's rings, where the thickness of the plate of air is considerable), and the resulting light is white. The phenomena of color may still, however, be produ- ced in thick plates, by superposing two of them in such a manner, that the wave which has the greater velocity in the first shall have the less in the second. We have only to place the plates with their principal sections per- pendicular or parallel, according as the crystals to which they belong are of the same, or of opposite denomina- tions. Thus, if both the crystals be positive, or both negative, they are to be placed with their principal sec- tions perpendicular; and on the other hand, these sec- tions should be parallel, when one of the crystals is po- sitive and the other negative. The reason of this is evident. 167. Let us now consider the effects produced when uniaxal crystal; Ei;. <_-ts produced when a polarized wave traverses a a polarized wave traverses a uniaxal crystal, in various directions inclined to the axis at small angles; and let us suppose, for more simplicity, that the crystalline plate is cut in a direction perpendicular to the axis. Let ABCD be the plate, and Ethe place of the eye. The visible por- tion of the emergent beam will form a cone, A EB, whose vertex coincides Fig. 107. ELEMENTS OF OPTICS. 357 with the place of the eye, and axis E 0, with the axis Eay coinciding of the crystal. The ray which traverses the crystal in wi'h the axis ** J ■ d undergoes no the direction of the axis, P 0E, will undrgo no change change; whatever; and will consequently be reflected or not from the analyzing plate, according as the plane of reflexion there coincides with, or is perpendicular to, the plane of first reflexion. But the other rays composing the cone will be modified in their passage through the crystal, and the changes which they will undergo will depend on their inclination to the optical axis, and on the position of the principal section with respect to the plane of pri- other rays win ... -i . ,. be modified. nntive polarization. Let the circle represent the sec- tion of the emergent cone of rays made by the surface A B of the crystal; and let MM' and N N', be two lines drawn through its centre at right angles, being the intersection of the same surface by the plane of primitive polariza- tion, and by the perpendicular plane, respectively. Now, the vi- brations which emerge at any point of these lines will not be resolved into two within the ^..v L. 41 1 Vibrations that crystal, nor will their places of polarization, that is, of win not bo vibration, be altered; because the principal section 0fresolved; JV— Section of the emergent pencil by the face of the crystal; Ffc. 109. Illustrations; 358 NATURAL PHILOSOPHY. Fig. 109. Illustrations; Vibrations that the crystal, for these vibrations, in the one case coincides resolved; with the plane of primitive polarization, and in the other is perpendicular to it. These waves, therefore, will be reflect- ed, or not, from the analyzer, according as the plane of reflexion there coincides with, or is perpendicular to, the plane of first reflexion. In the latter case, a black cross white or black will be displayed on the screen, and in the former a white cross- one. vibratiens that But the case is different with the vibrations which win be resolved; emerge at anv ^fam point, such as L. The principal section of the crystal for these vibrations, neither coincides with, nor is perpendicular to, the plane of primitive po- larization ; and consequent- ly the incident polarized wave will be resolved into two, within the crystal, whose planes of polariza- tion are respectively paral- lel and perpendicular to the principal section 0 L. The vibrations in these two waves are reduced to the same plane by means of the Reduced to the analyzer ; they will, therefore, interfere, and the extent same plane by . the analyzer, and oi that interference will depend upon their difference of interfere. phase. Fig. 107. ELEMENTS OF OPTICS. 359 Now, the difterence of phase of the two waves varies Extent of with the interval of retardation. When this interval islnterference ^ dependent on an odd multiple of half a wave length, the two waves difference of will be in complete discordance; and, on the other hand, phase* they will be in complete accordance, and will unite their strength, when the retardation is an even multiple of the same quantity. The successive dark and bright lines will, therefore, be arranged in circles. § 168. We have been speaking here of homogeneous Phenomena light. When white or compound light is used, the rin2:sProdllcedwitu to . . white light; of different colors will be partially superposed, and the result will be a series of iris-colored rings separated by dark intervals. All the phenomena, in fact, with the ex- ception of the cross, are similar to those of Newton's rings; and we now see that they are both cases of the same fertile principle,—the principle of interference. These rings are exhibited even in thick crystals, because the difference of the velocities of the two waves is very Analogous to small for rays slightly inclined to the optic axis. Newton's rings Fig. 110. Illustrations; § 169. We will now consider briefly the case of biaxal crystals. Let a plate of such a crystal be cut perpen- 360 NATURAL PHILOSOPHY. their fundamental property; Effects of biaxal dicularly to the line bisecting the optic axes, and let it crystals. })q interposed, as before, between the polarizer and ana- lyzer. In this case, the bright and dark bands will no longer be disposed in circles, as in tha former, but will form curves which are symmetrical with respect to the lines drawn from the eye in the direction of the two axes. The points of the same band are those for which the interval of retardation of the twTo waves, is constant. Lemniscatse and The curve formed by each band is the Lemniscata of James Bernouilli,—the fundamental property of which is, that the product of the radii vectores, drawn from any point to two fixed poles, is a constant quantity. The exactness of this law has been verified, in the most com- plete manner, by the measurements of Sir John Her- schel. The constant varies from one curve to another,— being proportional to the interval of retardation, and in- creasing, therefore, as the numbers of the natural series for the successive dark bands ; for different plates of the same substance, the constant varies inversely as the thick- ness. The form of the dark brushes, which cross the entire system of rings, is determined by the law which governs the planes of polarization of the emergent waves. It may be shown that two such dark curves, in general, pass through each pole; and that they are rectangular hyperbolast whose common centre is the middle point of the line which connects the projections of the two axes. Form of the dark brushes determined. END OF OPTICS. ✓ J'Jl22'«/7V^V ^P^.----,^&__3k