With the Compliments of Wm. TIarkness, 
TJ. 8. Naval Observatory, Washington, I). C. 
|From the American .Journal of Science and Arts, Vol. XYTTI, Sept.. 1879.] 
ON THE 
Color Correction of Achromatic Telescopes. 
By WM. IIAIiKNESS. [From the American Journal of Science and Arts, You. XVIII, Sept., 1879.] 
On the Color Correction of Achromatic Telescopes; 
by Wm. IIarkness. 
Although much has been written on the theory of the 
achromatic telescope, I am not aware that any attempt has 
hitherto been made to treat the color correction rigorously as 
a function of the wave length of the light; and, on that 
account, much obscurity, and some positive error, has crept 
into the text books on the subject. 
The theory given in the following pages is based upon funda- 
mental equations which neglect the thickness of the lenses, as 
has always been done heretofore, and which suffice to give the 
refractive indexes to scarcely more than four places of decimals. 
All the subsequent operations upon these equations are rigor- 
ously accurate; and it would have been useless to attempt 
greater precision in the refractive indexes, while the thickness 
of the lenses is neglected. As achromatic telescopic objectives 
are usually composed of two lenses, rarely of three, and hardly 
ever of a greater number; it has been thought sufficient to 
write the equations in the form applicable to triple objectives, 
but no difficulty will be experienced in extending them to a 
greater number of lenses, when necessary. 
Our fundamental equations are 
(1) 
m = a + hY* + cr4 (2) 
in which 
f — the principal focal distance of any lens. 
// the refractive index of the lens. 
r — the radius of curvature of the first surface of the lens. 
p — the radius of curvature of the second surface of the lens. 
A = the wave length of the light. 
y — l -7- A. 
<1, b, c = certain coefficients, determined from not less than three 
properly situated values of p. 
Equation (2) is Cauchy’s dispersion formula. Now put 
1 1 
1—- — A (3) 
r p v ' 
and suppose a series of lenses, such that 
= (/h -l)A.; j = (/b - !)■a,; j = (m»- 1)a, (4) 
These lenses being very thin, let them be placed in contact with 
each other; and let the equivalent focal distance of the whole 190 
W. Harkness—Color Correction of Achromatic Telescopes. 
combination be/ Then, by a well known optical theorem, 
J = (M1) A, + Ob- 1) A, + 1) A3 (5) 
Substituting the values of fxv //2, /xv from equation (2), 
putting 
C —A/a, —1) + A/a — 1) + A3(a3—1) 
D =A/ 4- AA -f* A/3 
E = A A + AA + A3C3 
(6) 
and arranging the terms according to the powers of f, we have 
f=C+T>y' + Ey* 
This equation expresses the relation between the focal dis- 
tance of the combination, and the wave length of the light. 
It shows that when white light enters an objective there will 
generally be an infinite number of foci, situated one behind 
the other, and all contained between the two values of / which 
correspond to the limiting values of y. For our purpose, how- 
ever, it will be more convenient to consider f as the ordinate, 
and y as the abscissa, of a curve which we will designate as the 
focal curve. To investigate its properties, we differentiate with 
respect to /and y, and obtain 
+ 2E/) (8) 
Putting the left hand member of this expression equal to 
zero, we find 
Differentiating (8) a second time 
0 = 2/y<2D + 4Er’)’-/’(2D + 12Er’) (10) 
Substituting the value of y2 from (9), this becomes 
f=4D/’ <n> 
which shows that, so long as D remains positive, the curve is 
convex toward the objective, and the value of y given by equa- 
(9) corresponds to the minimum focal distance. 
An achromatic objective, or more accurately, and with greater 
generality, a corrected objective, is one in which all rays of the 
kind for which the correction is made are brought to as nearly 
as possible the same focus. For example; if an objective is 
corrected for visual purposes, then the rays which produce the 
greatest effect upon the human eye must all be brought as 
nearly as possible to the same focus; or, if the objective is 
corrected for photographic purposes, then the rays which act W. liarkness—Color Correction of Achromatic Telescopes. 
191 
most energetically upon silver bromo-iodide must all be brought 
as nearly as possible to the same focus. This condition will 
evidently be fulfilled when the rays in question have the min- 
imum focal distance; or in other words, when they satisfy 
equation (9). Thus it appears that this equation determines 
the correction of the objective, and for that reason it will be 
called the achromatic equation, and the particular value of y 
which satisfies it will be designated as y0. 
To find the relative values of A„ Aa, A„ in a corrected 
objective, we substitute in (9) the values of D and E from (6). 
The resulting expression for the middle lens is 
a _ A,(ft, + 2c,?V) + As(63 + 2c3y0a) 
(*a + 2 c,r;) (12) 
which shows that this lens must be of the opposite kind from 
the other two,—that is, if the first and third lenses are convex, 
the middle one must be concave; or vice versa. 
To find the equivalent focal distance of the whole combina- 
tion for the ray /, (9) gives 
. D=-2Ey0a (13) 
Substituting this in (7) 
~ C-Ey04 
Replacing C and E by their values from (6) 
f'~A,(ocjV-l) +•&.(«, - +A,(a,-<yV—1) (15) 
Substituting the value of As from (12), and putting 
A3 
L= (a-1) (ft9+2c,^0s) - (a2-l)(*1H-2c,^0a) + y04(* c-b c,) 
m=(«3-!) (62+2(;2ro2)—K -!) (^3+2c3r02)+Yo - Kc>) 
(16) 
we obtain finally 
f_ K + 
-70 A,(L + wM) 1 
The ordinate of the focal curve for the ray yt„ is the difference 
between the focal lengths of the objective for the rays X0 and 
To find it we have 
J-J=(C+Dr.’+Er,,)-(C + Dr,-)+Er/)=D(/,*-ri-) + (Er,<-r,‘) (18) 
J 0 J 1 
1 1 f -f 
But (19) 
and putting/,—f0— Jf„ this becomes /^6E0l 
' ((20) 192 
W. Harkness—Color Correction of Achromatic Telescopes. 
Substituting for the quantity within the brackets, its value 
from (18) ; and replacing D and E by their equivalents from (6) 
| (A A + A A + A A){r'-hl+(A1c1 + AJcs+A 3c3)(y0*-r*) ((21) 
Substituting the values of A2 and A3-f- A, from (12) and (16), 
and putting 
N — b1c^-bj1 
P=^c,-Ac, (22) 
we obtain the important expression 
N «P 
=a jjxy: - r;y (23) 
If a star is viewed through a carefully focused achromatic 
telescope, and if the surface in the focus of the eye-piece is 
designated as the focal plane: then, of the infinite number of 
images which equation (7) shows will be formed, some will be 
situated before, and some behind the focal plane, but only one 
will coincide exactly with it. The cones of rays which form 
the images situated before and behind the focal plane will 
necessarily have a sensible diameter at their intersection with 
that plane, and their combined effect will be to produce a 
fringe of colored light around the image of the star, as seen 
through the ejm-piece. This fringe is the secondary spectrum, 
and its magnitude, for light of any given wave length, will 
evidently depend upon the value of Afv Hence, to destroy 
the secondary spectrum, Afx must be made equal to zero. 
Equation (28) shows that this will be the case for a triple 
objective when 
N -f- nF = 0 (24) 
or for a double objective when 
N = 0 (25) 
As yet no materials have been discovered whose physical 
properties are such as to satisfy these conditions. We there- 
fore proceed to investigate what form an objective constructed 
of any given materials must have in order to render the sec- 
ondary spectrum a minimum. 
Substituting in (28) the value of A, from (17), we find 
<»> 
In the right hand member of this equation, n is the only 
quantity which depends upon the form of the objective. Con- 
sidering it as variable, and differentiating, we obtain 
dW=«r:-r,rfcSjr ; <*> 
To make J/j a minimum, such a yalue must be attributed to 
n as will reduce the right hand member of (27) to zero. This W. I lark ness—Color Correction of Achromatic Telescopes. 
193 
condition gives at once, n=oo; which will be the case when r 
and /> are both infinite; as is evident from equations (16) and 
(3). The objective is then reduced to two lenses, and a piece 
of very thin piano-parallel glass. As the latter cannot appre- 
ciably affect the color correction, it may be dismissed from 
further consideration; and thus it appears that from any three 
pieces of glass suitable for making an objective, but not ful- 
filling the conditions necessary for the complete destruction of 
the secondary spectrum, it will always be possible to select two 
pieces from which a double objective can be made that will be 
superior to any triple objective made from all three of the 
pieces. 
The focal curve being tangent to the focal plane at the point 
corresponding to the wave length ; if we assume the spherical 
aberration to be perfectly corrected for light of all degrees of 
refrangibilitv; then the image of a star formed upon the focal 
plane by light of wave length A0 will be a point, and the linear 
semi-diameter of the image of the same star formed by light of 
wave length will be the semi-diameter of the cone of rays of 
that wave length at the point where it cuts the focal plane. 
Therefore we have 
Afx: s0li (28) 
in which a is the semi-aperture of the objective, and s0n is the 
required semi-diameter of the cone of rays of wave length 
Combining (28) with (26), we find 
5ou = <*(r: - y?y w 
This is the expression for a triple objective. In the case of 
a double one, n becomes zero, and (29) reduces to 
C = a(Yo' - Y'y (30) 
which shows that in a double objective properly corrected for 
any given purpose, the linear semi-diameter of the secondary 
spectrum is absolutely independent, both of the focal length of 
the combination, and of the curves of its lenses; and depends 
solely upon the aperture of the combination, and the physical 
properties of the materials composing it. 
If a telescope armed with an achromatic eye-piece is carefully 
focused upon a star, and then the image of the star is viewed 
through a prism held before the eye-piece; it will be seen that 
the eye does not adjust the focal plane tangent to the focal 
curve, but places it somewhat further from the objective, in 
such wise that the plane cuts the curve in two points, which 
we will designate as ym and yn. For these points we must have 
C + Dr.’ + Ey.‘= 0 + Dr: t Er.‘ (31) 194 
W. Harkness—Color Correction of Achromatic Telescopes. 
which gives 
“ E = r~ + r* (32) 
But by (9) we have 
r'=-m <33> 
Combining this with (32), we find 
r; = i(yj + y„°) (34) 
which gives the relation between y0 and any pair of points at 
which the focal plane may cut the focal curve. 
We have next to consider how the value of ym can be found ; 
and for that purpose a method partly arithmetical, and partly 
graphical, seems most convenient. The data required are, the 
values of Af for a number of different values of y, and the 
relative intensity of the light at each of these values of y. 
The values of Af must be computed by means of equation 
(26); and the relative intensity of the light may either be 
determined experimentally, or taken from published tables. 
For visual intensity, the table given by Fraunhofer may be 
employed; and for photographic intensity, the curves pub- 
lished by Captain Abney contain all that is required. For the 
sake of definiteness, let us suppose that the value of ym is to 
be determined for an objective corrected for visual purposes. 
We begin by laying down an axis of abscissas, and graduating 
it into a scale of wave lengths. Here, however, it must be 
observed that the brightness of any part of a spectrum depends 
not only upon the inherent brightness of the light at that 
point, but also upon the degree of dispersion employed. As 
Fraunhofer’s determinations of the relative brightness of dif- 
ferent parts of the spectrum were made with a flint glass prism 
having a refractive index of 1/63 for the ray D ; and as such 
an instrument produces much greater dispersion at the violet 
end of the spectrum than at the red end; it follows that our 
scale of wave lengths must be, not a scale of equal parts, but 
such a scale as existed in the spectrum employed by Fraunhofer. 
The wave length of the brightest ray is approximately 5688, 
and through that point in the scale, and at right angles to the 
axis of abscissas, the axis of ordinates must be drawn. Then, 
from the computed values of Af a sufficient number of points 
must be laid down to determine the focal curve, and that curve 
must be drawn. At the points whose wave lengths correspond 
to the principal Fraunhofer lines, lines must be drawn through 
the focal curve, parallel to the axis of ordinates; the length of 
each line being proportional to the relative brightness of the 
spectrum at the point where it is situated, and the center of 
each line coinciding accurately with the focal curve. Through W. Harhiess—Color Correction of Achromatic Telescopes. 
195 
the extremities of these lines a closed curve must be drawn. 
The figure thus obtained will be termed the illumination dia- 
gram, because it exhibits the amount and distribution of the 
light at the focus of the objective. The eye will necessarily 
place the focal plane in the position where this light will pro- 
duce the greatest effect upon the retina; which is equivalent 
to saying that the focal plane must pass through the center of 
gravity of the diagram. Hence, to find the position of the 
focal plane, we have only to cut out the diagram (which should 
be drawn upon rather stiff paper), and balance it upon a knife 
edge held parallel to the axis of abscissas. The reciprocals of 
the wave lengths of the points of intersection of the knife edge 
with the focal curve will then be the values of ym and yn. 
The method just explained may be employed to determine 
the difference between the positions of the principal focus of 
the same telescope when used for different purposes. For 
example, if it were required to find the interval between the 
visual and photographic foci of a telescope, two illumination 
diagrams would be drawn—one for the visual, and the other 
for the photographic rays—and the difference between the 
positions of the focal plane in the two diagrams would be the 
required difference of foci. 
As the magnitude of the secondary spectrum of a star is 
measured by the semi-diameter (at the point where it intersects 
the focal plane) of the cone of rays having the maximum focal 
distance; it follows that in an objective corrected for visual 
purposes, the secondary spectrum is diminished by the fact 
that the eye places the focal plane somewhat further from the 
objective than the apex of the focal curve. To find the 
amount of this diminution, we remark that for light of wave 
lengths corresponding to the points where the focal plane cuts 
the focal curve, the semi-diameter of the cone of rays is zero; 
while for light of any other wave length, the semi-diameter of 
the cone of rays, at the point where it intersects the focal plane, 
is proportional to the distance between that plane and the point 
of the focal curve corresponding to the wave length of the light. 
Hence, the effect of moving the focal plane into a position 
further from the objective than the apex of the focal curve, 
will be to diminish s0n by a constant which is numerically 
equal to the value of s0n for light whose wave length is that of 
the point at which the focal plane intersects the focal curve. 
Modifying equation (30) in accordance with these principles, it 
becomes 
= a{ (yo2 - y'Y - (yo2 - ym'Y\ (35) 
in which ym is the reciprocal of the wave length corresponding 
to either of the two points in which the focal plane cuts the 196 
W. IlarJcness—Color Correction of Achromatic Telescopes. 
focal curve; and smu is the semi-diameter, at the point where it 
cuts the focal plane, of the cone of rays whose wave length is 
The exact nature of the color correction of a telescope can be 
determined by placing the focal plane in a number of different 
positions, and observing the corresponding values of ym and yn. 
These values being substituted in equation (34), several inde- 
pendent values of y0 can be deduced, the mean of which will 
probably be very near the truth. 
The conclusions reached in the preceding pages may be 
summed up as follows: 
1st. From any three pieces of glass suitable for making a 
corrected objective, but not fulfilling the conditions necessary 
for the complete destruction of the secondary spectrum, it will 
always be possible to select two pieces from which a double 
objective can be made that will be superior to any triple 
objective made from all three of the pieces. 
2d. The color correction of an objective is completely defined 
by stating the wave length of the light for which it gives the 
minimum focal distance. 
3d. An objective is properly corrected for any given purpose 
when its minimum focal distance corresponds to rays of the 
wave length which is most efficient for that purpose. For 
example, in an objective corrected for visual purposes the rays 
which seem brightest to the human eye should have the mini- 
mum focal distance; while in an objective intended for photo- 
graphic purposes the rays which act most intensely upon silver 
bromo-iodide should have the minimum focal distance. 
4th. In double achromatic objectives the secondary spectrum 
(or in other words, the diameter, at its intersection with the 
focal plane, of the cone of rays having the maximum focal 
distance), is absolutely independent both of the focal length of 
the combination, and of the curves of its lenses; and depends 
solely upon the aperture of the combination, and the physical 
properties of the materials composing it. 
5th. When the focal curve of an objective is known ; and 
the relative intensity, for the purpose for which the objective 
is corrected, of light of every wave length is also known; then 
the exact position which the focal plane should occupy can 
readily be calculated. 
6th. It may be remarked incidentally that in an objective 
corrected for photographic purposes, the interval between the 
maximum and minimum focal distances is less than in one 
corrected for visual purposes. Hence, a photographic objective 
has less secondary spectrum, and is better adapted to spectro- 
scopic work, than a visual objective. 
Washington, May 24. 1879.