A NEW CENTER OF GRAVITY FORMULA OF GENERAL 
APPLICABILITY 
By J. WOODBRIDGE DAVIS, O.E. 
Written for Van Nostrand’s Magazine. 
The formula 
ip+(B-r. (i> 
The distance, D3, of the center of gravity 
of the two spaces from the initial plane, 
is ' 
p_P,V, + P,V,- 
3~ v1+y2 
jPV, + tV(Bi-A1)D’1 + jDV, + -A,)P' 
V, + Va 
— ID _L (Pi _lp I (^8~^-3)P* 
12(Y,+Va) 12VS 
wherein A is the area of the generatrix 
of a space, in its initial position, B, its 
area in its final position, D is the distance 
between these positions, and Y is the 
amount of space between, is an expres- 
sion representing a distance, or linear 
measurement, like in kind to D, because 
the second term is a product of four 
dimensions divided by a product of three 
dimensions of like kind. If the space be 
a plane, the second term is still, as well 
as the first, an expression of a single 
dimension. This expression has been 
found to represent the distance of the 
center of magnitude from the initial end, 
of each of a number of shapes, or the 
center of gravity of a mass of uniform 
density, which fills the space.* 
To determine the extent of its applica- 
bility, let us first consider twro spaces, 
wherein As, B3, Vs, are the areas of ends, 
and volume of combined space. There- 
fore, formula (1) applies to the space 
y=L/» +/,(*)].=/,(*)„• 
In the same manner it may be shown 
that the formula applies to the space 
y =[/»+/»]«, 
D 
if it apply to space, y— fXx)0> and so on. 
Consequently, if formula (1) apply to 
each of any number of spaces between 
same limits, it applies to their sum. 
From a similar course of reasoning it 
follows that If formula, (1) apply to all 
but one of any number of spaces be- 
tween same limits, it does not apply to 
their sum. Also, if the formida apply 
to certain spaces between same limits and 
do not apply to certain other spaces, more 
than one, between the same limits as 
before, the formula does not apply to the 
sum of all the spaces, except in special 
cases, when its error for some of the 
spaces is balanced by its error, with 
opposite sign, for the rest. 
Now let us consider the expression for 
y =/>)„, y=f,ix) o- (2) 
Let Aj, Bj, be the end areas of first, V,, 
its volume, D, the distance between these 
limits, and D,, the distance, from first end, 
of the center of its magnitude. Let A2, 
B2, Y2, D, D2, be the similar values of the 
second space, both lying between same 
limiting planes, as indicated in (2). 
If formula (1) apply to each of these 
spaces, then 
D =£D+ 
1 * 12Yj ’ 
D =iD + A^pa 
'J * + 12Y4 * 
* Formulae forR. R. Earthwork, Second Edition, p. 105. 28 
A NEW CENTER OF GRAVITY FORMULA. 
all spaces, ?/=F(a:).* Each term of this 
is of the form 'Kxn; and the distance of 
the center of gravity, from the first end, 
of the space represented by y=Kxn, is 
position whatever, since to shift the axis 
of Y does not change the degree of eq. (5), 
is all that requires demonstration. The 
position of the center of magnitude in 
that space is, then, immediately indi- 
cated by formula (1), which may be 
reduced for particular cases. 
For instances: Because the areas of 
generatrices of triangle, pyramid, cone, 
and paraboloid, in terms of distance 
from their vertices, are, respectively, ax, 
b(cx)3, 7t(exy ancl nfx, formula (1) 
applies to all of these and to all of their 
frusta. Therefore, for triangle, the dis- 
tance of center of gravity from vertex is 
/D 
, ldx w+i 
/■“m “ " ' - 
»/ 0 
Formula (1) applies to this space when 
— KDn+i 
w + 1 
This is true when ra=0. For all other 
values of w, (3) becomes 
l (B-O)D3. 
D+ 12 (£DB) “ D‘ 
P + (4) 
12 n + 2 
From this n*—3n + 2=0; 
whence n—f±£=2 or 1. 
For trapezoid; 
v , f lr> - (B-A)D’ (2B-A)D 
dist. c. oig.=^+im^m= ~3(B + A)* 
For pyramid or cone when B is area of 
base, 
It follows from this and the rules 
written in italics, that the center of 
gravity formula applies, between any 
limits, to those spaces only which are 
represented by the equation 
disto.ofg.=iD + i^^.)=}D. 
For frustum of this: 
y=a + bx + cxt; (5) 
dist. c. of + = 
B 2 12(A+a/AB+B)!D 
£D + (B-^)D— 
4(A + a/AB + B) 
and it applies to special cases only, 
[i. e., between special limits,] of all other 
spaces. 
By comparison of (5) with similar 
equation, [eq. (11), p. 412, May No. of 
this Magazine,] representing the limit of 
the prismoidal formula’s applicability, 
which equation is 
The last result, which is formula (1) 
only changed by cancelation of the com- 
mon factor D, is at once in the simplest 
form. Dr. Weisbach requires a page 
and a quarter of laborious demonstration 
to reach this result. See pp. 233-4, Eck- 
ley Coxe’s edition of Theoretical Me- 
chanics. 
For paraboloid, the radius of whose 
base is r, 
y=a + bx + ex* + cfa3, (6) 
it is seen that the center of gravity 
formula is, practically, co-extensive with 
the prismoidal formula; because the 
generatrices of very few, if any, practical 
shapes vary as cubic functions of the 
path. This formula will, therefore, 
serve in the same way as the prismoidal, 
as a widely general rale, which renders 
mmecessary the demonstration, recollec- 
tion and use of a large number of special 
rules. 
Thus, the chapter on center of gravity 
in a treatise on mechanics can be much 
abbreviated by use of this formula. The 
fact that the generatrix of a space, ex- 
pressed in terms of its distance from 
initial end of that space, or from any 
dist.c.ofg.=iD + i|^B=|D. 
For frustum of this we obtain an ex- 
pression similar to that for trapezoid, 
since its generatrix varies according to a 
function exactly similar. It is 
7r(r'—r a)D5 
diet. c. of g. = £D + ; . 4Trh = 
& 127r(r, H-r, )£D 
(^-QP 
3(r* + r’) 
Formula (1) applies to the sphere, 
because, with center as origin, the mag- 
nitude of generating circle varies as 
* See introduction to article entitled Prismoidal For- 
mula in May number of this Magazine. A NEW CENTER OF GRAVITY FORMULA. 
29 
7r(r2 —x2). 
Fig. I. 
Hence, for hemisphere, measured from 
center, 
7tr* 
dkt. o. ofg.=ir—=|r. 
Because the generatrix of the complete 
rectangular prismoid [Fig. 2, p. 414, May 
No. of this Magazine] varies as 
a + bx + ccc2, 
formula (1) applies to all rectangular 
prismoids and wedges, in direction per- 
pendicular to bases. 
. * 1T. , (B—A)D2 
tet. c. of g. =iD + 12(1+4MVb)|I>= 
1 B-A 1 (B + 2M)D . 
* } A + 4M + Bf-A + 4M+B 
For wedge, measured from edge; 
dist.c.ofg.=iDjl+g|ai| 
_(B + 2M)D 
B + 4M 
about CX, then the circumference~of the 
generating circle whose radius is y, gen- 
erates in the time, dt, 
«r--i 
To reach results similar to these for 
wedge and prismoid requires more than 
a page of Weisbach’s Mechanics. By 
calculus, the demonstration is nearly as 
long. 
In a manner similar and quite as easy 
it may be shown that formula (1). applies 
to any segment of common parabola, in 
direction perpendicular to axis, to all 
parallelograms, prisms, cylinders, pris- 
moids, eylindroids, spheroids, hyperbo- 
loids, all segments of these between 
parallel planes, and to the class of shapes 
whose lateral boundaries are right line 
surfaces the type of which is illustrated 
in Fig. 6, p. 418, of May No. of this 
Magazine. The last class includes alone, 
columns, chimneys, piers, abutments, 
warped-faced wing-walls, banks, retain- 
ing walls, dams, earthwork solids, and 
numerous other practical shapes. 
If the extremity, S, of the generatrix 
of a circle, Fig. 1, generate a length, dx, 
of the circumference in an infinitesimal 
time, dt, while it passes through center, 
then, at any distance, x, it will in same 
time generate a length equal to 
V 
dx. -.2 Tty= 2 nr dx. 
y J 
which is the form of eq. (5), when b—o, 
c—o. Hence formula (1) applies to any 
zone of sphere, and the position of cen- 
ter of magnitude is, when h is altitude of 
zone, 
, j i (2nrdx-2nrdx)h?7 
2 fi 4 12~V 2a * 
Because formula (1) applies to the 
cone CSS', and to the segment SXS', as 
has already been shown, it may be used 
to determine the positions of the centers 
of gravity of these; and the c. of g. of 
sector may then be found by composi- 
tion of moments. 
Formula (1) applies directly to the 
spherical sector when D is understood 
to be the distance passed over by the 
center of magnitude of generatrix, a 
varying, concentric zone, and V to be the 
amount of space described by a plane 
generatrix, varying as the same function 
of, and passing over, the same path. 
This is true, because the distribution 
of magnitude along the path D is identi- 
cal for both modes of generation, and 
because, since formula (1) applies to one 
of these shapes, the variation of genera- 
trix being as ex2, it also applies to the 
other. 
Thus is obtained for sector of sphere, 
ST v r 
dx. =,TT=dx.-=clx. —; — 
SU y yV-s«2’ 
Hence formula (1) does not apply to 
arcs of circles. 
But, if the circumference be revolved 30 
A NEW CENTER OF GRAVITY FORMULA. 
dist. c. + 12^)=f^ 
=f HA) • • (7) 
y=\bx Jx*_ _1 -J_ nap. log. 
Y 2 b 
| x+ |/a;2— J_ | + const. 
where B is area of spherical circle, d is 
distance of its c. of g. from center of 
sphere, h is its altitude, and r is radius 
of sphere. 
In this sense formula (1) applies to, 
frusta of spherical sectors, [i. e., be- 
tween concentric zones], whether the 
zones be terminal or annular; also, in di- 
rection of sector’s axis, to all divisions of 
these included each by two planes passed 
through axis; to circular sectors between 
any limits, to all portions of regular 
polygons and polyhedrons, which can be 
inscribed in circular and spherical sec- 
tors, respectively, between any limits, 
and to every shape, each part of whose 
generatrix varies as the same function of 
its distance from initial end. 
Thus, it applies to the square diago- 
nally, when the initial end is one vertex, 
and the generatrix is the broken line 
composed of the opposite two sides. 
Applied in this way, formula (1) gene- 
rally requires, preliminarily, the aid of 
some other method to determine the c. j 
of g. of generatrix itself. In case of 
spherical sector, it applies both to the 
generatrix and to the space described 
thereby. 
It will be observed from the foregoing 
that the only familiar simple shapes, 
such as are made examples in works on 
mechanics and engineering, to which 
formula (1) does not apply directly, are 
the circular arc, sector, segment and 
spandrel; and to latter three of these it 
applies either indirectly or with aid of 
composition of moments. 
Formula (1) applies to all lines for 
which 
Formula (1) applies in direction of 
axis to all surfaces formed by revolution 
of the lines for which 
y +l=a + bx + cxi, (9) 
where a, b, c, are arbitrary. The only 
familiar surfaces of revolution for which 
eq. (9) is true are the surfaces of cylin- 
der, cone and sphere. 
Formula (1) applies, as a center of 
pressure formula, to the common cases of 
hydrostatic pressure. The equation of any 
immersed plane figure, referred to inter- 
section of its plane with surface of fluid, 
as the axis of Y, and to any line in its 
plane, perpendicular to this intersection, 
as the axis of X, is 
If 6 be the angle of deviation of its 
plane from plane of fluid surface, and y 
be the specific gravity of the fluid, sup- 
posed to be uniform in density, the press- 
ure on the figure at a distance x is 
2/=F(a;). 
y'=F(x)y sin 6 x=F (x). 
In order that formula (1) shall apply, 
F '(x) — a + bx + ex'. 
-r*i v a + bx + cx2 a' ,, , 1A 
F(ar) = 7—5—=—+b +c«.(10) 
y sin U.x x 
Fortunately, the last member of eq. 
(10), when we make af=o, represents 
nearly all the usual submerged plane 
shapes, since these are seldom other 
than the rectangle, triangle and trape- 
zoid, with bases parallel to fluid surface. 
Because the pressure at every point of 
submerged surface is normal thereto, 
and equal in intensity to y sin 6 x, it 
may be represented by a line perpendic- 
ular to surface and varying in length as 
the same constant multiple of x. There- 
fore, the equation of entire pressure, 
y + \—a + bx-\- cx*, (8) 
dx 
where a, b, c, are arbitrary. If c=o, dy 
is always in integrable shape; but the in- 
tegration is tedious. Eq. (8), however, 
shows at once that the only familiar line 
to which formula (1) applies is the 
straight line. To obtain another ex- 
ample, without much labor in the inte- 
gration, make a and c zero and square, 
subtract unity, extract square root, and 
multiply by dx, each member of eq. (8). 
By integration the equation of a line, 
subject to formula (1) is found to be 
y=(b' + c'x)y sin 6. x=bx + cx*, 
is also the equation of a space, whose 
generatrix is a rectangle varying as y in 
last equation, its sides remaining con- 
stant in direction. This shape can, con- 
sequently, be only a frustum of the rec- 
tangular pyramid, prism or prismoid. A NEW CENTER OF GRAVITY FORMULA. 
31 
We may, therefore, confine our attention 
wholly to these shapes, because the re- 
sultant of pressure passes through the 
center of magnitude of each. 
Of course, as shown by expression (6), 
the prismoidal formula also applies to 
these shapes, and, in consequence, to the 
pressure. Indeed, it would seem that 
this is the simplest method of determin- 
ing the total pressure. The following 
is the 
KULE. 
To find the total pressure on a sub- 
merged rectangle, triangle or trapezoid, 
whose bases are parallel to fluid surface, 
Multiply top-width of shape by its dis- 
tance beneath surface ; call this A. Mul- 
tiply bottom width by its distance beneath 
surface; call this B. Multiply mid- 
width by its distance beneath surface / 
call this M. Then total pressure is of 
distance between end-widths, multiplied 
by the specific gravity of fluid, and 
again by 
and formula (13a) becomes 
a , ip | (v,d,-"A)p 
1 2 2 [yold1 + (w1 + ?c2)(d1 + d2) + wfij 
(15) 
as simple as before. 
To find distance of center of pressure 
below fluid surface, multiply (15) by sin 6, 
or use in (15), instead of diy c?2, D, their 
vertical projections, if these be known 
EXAMPLES. 
The submerged figure is a triangle 
whose base is at fluid surface. Here, dx 
and w2 are zero; consequently, the dis- 
tance of center of pressure is £D. The 
pressure is 
T)y sin 6 wxd2 or D y 
This corresponds to the case of the 
entire middle branch of the complete 
prismoid, Fig. 2, p. 414, May No. of this 
Magazine. 
The vertex of the triangle is at sur- 
face, and the base is parallel thereto. 
Here wx and dx are zero; and, in conse- 
quence, formula (15) reduces to £ D. 
Then, 
A + 4M + B. 
When top-width is w2, and its dis- 
tance beneath surface is h2; and w>2, A2, cor- 
respond to lower width, while D is dis- 
tance between, the formula is 
sin 6. wfi2, or JDy wji2. 
p=JT>VOA + (m, + to,) (A, + h ) + wfij 
• • (11) 
The figure is a rectangle. Here wx = w2, 
and the formula (15) becomes 
If it be desired to determine the com- 
ponent of pressure in any direction, use, 
instead of I), the projection of D normal 
to that direction. 
The formula determining the distance 
of center of pressure, measured from fluid 
surface along submerged plane, is 
1 5 T 6(c?a + c?j) 
sin 6. 
The distribution of pressure on sub- 
merged trapezoids corresponds to the 
distribution of magnitude in the various 
segments of the complete rectangular 
prismoid, as wedges, etc. 
If the plane of submerged figure be 
parallel to fluid surface, formula (13a) 
shows that the center of pressure is co- 
incident with the center of magnitude of 
the plane shape itself. When, now, 
— we have the case corresponding 
to that of the rectangular prism. 
While defining the rectangular pris- 
moid on p. 414, May number of this 
magazine, as a shape generated by a 
moving rectangle, the product and quo- 
tient of whose two dimensions vary, we 
noticed two other shapes, in one of 
which—the pyramid—the product is va- 
riable and the quotient constant, and in 
the other of which—the prism—both are 
constant; also, a fourth shape was de- 
+ ■ . (12) 
where is distance from fluid surface to 
top base, measured on plane of figure, 
and the other symbols are as before. 
(12) may be written 
cZi + 2D + 2(B+4M+A) ’ (13) 
or 
d +iD+ fej.-y,A)D 
’ 2[w?1A1 + (w1 + wa)(A1+A2) + aAJ 
. . . (13a) 
When, instead of h2, A2, we know the 
distances d2, d2, along submerged plane, 
formula (11) becomes 
P=iDy + {w 4- wj (d1 + d2) 
+ ™A] ■ ■ • (14) 32 
A NEW CENTER OF GRAVITY FORMULA. 
scribed for which the product is constant 
and the quotient variable. All these 
cases occur in hydrostatics. The only 
one not mentioned is that of the figure 
whose generatrix varies in magnitude as 
Qj 
—, the first term of last member of eq. 
x 
(10), while the pressure varies as cinx. 
Formula- (13), or its equivalent, formu- 
la (15), in same manner as formula (1), 
saves the calculator the inconvenience of 
remembering numerous rules, because 
the reduction for special cases, a few of 
which have been illustrated, can be in- 
stantly effected before he proceeds to 
the numerical part of the work. The 
same may be said of the prismoidal for- 
mula, as applied to the summation of 
these pressures. 
The only other submerged figure men- 
tioned in treatises on hydrostatics is the 
circle, which is usually the surface of a 
valve. Formula (15) does not apply to 
this. If r be its radius, and h the depth 
of centre below surface, then 
To find the moment of the horizontal 
component of P, referred to fluid sur- 
face, which is the moment usually re- 
quired, multiply the former moment by 
(sin Oy. 
This is useful when we would find the 
position of center of pressure of a com- 
pound figure. Such a figure may be divi- 
ded into triangles, trapezoids and rect- 
angles, whose bases are parallel to fluid 
surface; and the moment of each may be 
found as above. 
The moment which is most often re- 
quired is that of the horizontal compon- 
ent of P, referred to the lower base. 
Tins is the product of sin 6. P and sin 6. 
[ds— (15)], which is 
£Dsin'2 6. P — &2y sin8 6. (wid-wldl), or 
*6.y[2W'dl + (to, + wt) (dt + ]. 
(17) 
This is the moment which tends to 
overturn the solid whose surface receives 
the pressure. 
Formula (1) will prove useful to the 
practical engineer, since few shapes 
to which it is inapplicable come under 
his consideration. In his service i$ will 
be especially simple, for the reason that 
he will be very likely to have already 
calculated,_ for other purposes, the con- 
tents of the shapes he deals with, and 
will, consequently, know at the outset the 
value of the denominator of second 
term. For instance, as often occurs, if 
it be required to find the position of the 
c. of g. of a piece of iron or timber of 
known volume or weight and of pris- 
moidal shape, for the purpose of hoisting 
it, loading it upon a vehicle, or because 
it is a member of a structure or machine, 
the formula 
P =hitifiy\ 
r2 
and dist. cen. of press. —h + ~. 
The statical moment of a material 
shape, represented by eq. (5), referred to 
an axis parallel to plane of generatrix, 
and at a distance beyond its initial 
position, is the product of V by the sum 
of dx and expression (1). 
. •; Statical moment = (dx 4- |D)V + J^D2 
(B-A). ’ (16) 
This has exactly same advantages as (1). 
Formula (16) is very convenient when 
we would determine the center of gravity, 
or the statical moment of a compound 
figure, every part of which is a shape 
represented by eq. (5). The first term of 
second member shows that the c. of g. of 
each part may be assumed to be half way 
between its ends; whereafter, by compo- 
sition, very easily a false statical moment 
of the whole figure can be obtained. 
This should be corrected by the second 
term of second member applied to each 
part. 
The statical moment of the pressure, 
whose resultant occupies the position 
indicated by (15), is, referred to intersec- 
tion of submerged plane and fluid sur- 
face, the product of P and expression 
(15). 
(B-A)D2 
12V ’ 
indicating its distance from mid-section 
toward larger end, is exceedingly simple. 
If y be the specific gravity of the mate- 
rial and W the weight of piece, the ex- 
pression becomes 
(B-A)Dy 
12 W 
Even when neither the weight nor 
volume are known, the practical calcu- 
lator will find formula (1) very conveni- 
ent, because it can so easily be reduced 
to the simplest possible form for special A NEW CENTER OF GRAVITY FORMULA. 
33 
cases, by mere cancelation of symbols, 
before the numerical part of the work be 
commenced. 
The calculator can, in most cases, re- 
cognize at sight the shapes to "which for- 
mula (1) applies. For doubtful cases 
formula (5) is the criterion, and it is 
easily used. When the shape satisfies 
this, it is known that both the prismoidal 
and the center of gravity formulae apply. 
The best practical application of for- 
mula (1) is to the determination of the 
mean distance, which the material of a 
cutting has been hauled to form an em- 
bankment, since here it apparently satis- 
fies the greatest want. Haulage is the 
product of the quantity of material and 
the average haul or mean distance which 
it has been transported. The unit of 
haulage is one cubic yard haided one 
hundred feet. On this basis the price 
for haulage is fixed. 
After having calculated one factor of 
haulage, the quantity, it remains for us 
to find the other, or average haul. To 
do this pass a plane anywhere between 
cut and fill normal to route of haul. 
Suppose the magnitude of cutting to be 
generated by a limited plane, whose vari- 
able area is represented by y. Let x de- 
note the variable distance of y from the 
secant plane. However irregular may 
be the shape of cutting, we know that y 
varies as a function of x. Hence ydx, 
the elementary volume, multiplied by x, 
which produces the elementary amount 
of haulage, is integrable. 
the average haul from same plane to fill 
is equal to the distance of c. of g. of fill 
from plane. 
Therefore, the average haul of a piece 
of excavation is the distance between the 
center of gravity of the material as found 
and its center of gravity as deposited. 
This is as it is stated in works which 
touch upon the subject. But the prac- 
tical computation of this theoretical re- 
sult has been found to be a far more te- 
dious task. It is evident that, first, the 
centers of gravity of the component 
solids must be severally ascertained, 
since the cut or bank is measured as a 
compound shape. But the application 
of formula (18) to each of these produces 
a very intricate expression, involving 
about double the labor necessary to cal- 
culate the true content of the solid by 
means of the prismoidal formula in 
crudest shape. After this the several 
moments must be compounded. 
To avoid this some calculators have 
been in the habit of dividing the excava- 
tion into two parts of equal volume by a 
plane normal to center line, and estab- 
lishing this as the initial point of the 
average haul. A plane similarly fixed in 
the embankment marks the terminal 
point of same distance. But this plane 
is always nearer the larger end of shape 
than the c. of g. is, as may be illustrated 
upon the cone, triangle or any shape of 
unequal end dimensions. 
For instance, if the first five 100 ft. sol- 
ids of a railroad cutting have been trans- 
ported to a bank, and the generatrix 
commence with an area zero at beginning 
of cut, and reach, at the end of consid- 
ered part, an area whose center height 
is 20 ft., road bed width 20 ft. and side 
slopes 1\ to 1, the total volume is about 
10,000 cubic yards, and, consequently, a 
difference of 1 ft. in distance makes a 
difference of one dollar in money. But 
the difference in the cut is 20 ft., and, if 
the bank be of same form, 20 ft. is there 
added to average haul distance. The 
average haul of other end of same cutting 
is likely to be also 40 ft. too long. The 
error of this method, then, makes a total 
error of eighty dollars for that cutting, 
which is invariably at the expense of the 
railroad company. In the time that a 
calculator would, by this method, com- 
pute the haulage of a division of ten 
miles he would be likely to cost his em 
/id 
J * xydx, 
where c?2, are distances of ends of cut- 
ting, is the haulage of cut to plane, and, 
when x' is the average haul so far as 
to the plane, the following equation is 
true. 
fd'1 xydx 
x- X—• (18) 
J ch Vdx 
The first member of this equation is, 
by definition, the average haul to plane; 
the second member is, by principles of 
mechanics, the distance of c. of g. of 
material from plane, and the equation 
shows that these are equal. 
By similar reasoning it is proven that 34 
A NEW CENTER OF GRAVITY FORMULA. 
ployers an amount equal to a year’s 
salary. 
To divide the cutting into numerous 
small parts, and find the sum of their 
moments, or to determine the c. of g. 
by a mental estimate merely, are methods 
either laborious or liable to error. 
Because the earthwork solid belongs 
to the class of shapes bounded laterally 
by straight line surfaces, formula (1) ap- 
plies thereto. Let A, B, C, . . . . K, 
be the areas of cross-sections, a constant 
distance apart, 100 feet, or D, in a piece 
of railroad excavation, whose material is 
all carried in same direction to form an 
embankment. Assume an axis, outside 
of cutting, a distance rf,, equal to 50 feet, 
beyond A. Then, according to formula 
(16), the statical moment of first volume, 
referred to this axis, or the haulage of 
its material to this axis, is, since d + 
=100, 
Sage }=V* + *V<*+ . . . . nVoln 
+«f(K-A) (21) 
Obviously, the axis may be established 
anywhere. It is merely convenient to 
place it midway between the 100 ft. sta- 
tions. It might occupy the position 
midway between A and B. Then the 
moment, or haulage thereto, would be 
expressed by formula (21), with each co- 
efficient of all but final term [called the 
correction term] decreased by unity. So 
the first term, Volx, would vanish. 
Since this is so, the embankment can 
be referred to the same axis, wherever 
that may be with respect to the bank. 
In short, all the terms, except the correc- 
tion term, express the operation, to mul- 
tiply the number of cubic yards in each 
volume by the number of hundred feet 
that the mid section of that volume is re- 
moved from the axis. 
It is well to place the axis between 
cut and fill, or, if they overlap, as nearly 
so as possible, in order to avoid negative 
moments or haulage. For instance, the 
axis might be established half way be- 
tween B and C. Then the partial haul- 
age would be 
100V1+*D’(B—A), (19) 
The haulage of second volume is, evi- 
dently, 
The haulage of each remaining volume is 
represented by a similar expression, ex- 
cept that the coefficient of first term is 
always the product of 100, or D, by the 
ordinal number of the volume. Thus, 
for last, or nth, volume, the haulage is 
200 V2+^D2(C—B). (19a) 
-Vol1+Vola + 2Voli . . . + (n-2)Voln 
+ (K~A>' <22> 
100 n V„ + TV D*(K—J) (196) 
This is less than (21), but the difference 
in defect, 2 Yol, where Yol is total vol- 
ume in cubic yards of material removed, 
is exactly balanced by the same differ- 
ence in excess, which is created when 
the haulage from axis to fill is consid- 
ered. 
If the irregularity of the ground sur- 
face make it requisite to sub-divide one 
or more of the 100 ft. volumes, as the 
volume between H and I in Fig. 2, where, 
also, a plus or intermediate station, K', 
instead of a full station, terminates the 
portion of cutting carried one way; the 
same rule expressed in italics, second 
paragraph above, holds, but the correc- 
tion formula is, for example above, 
The sum of these expressions is the total 
haulage to the axis. But the sum of all 
the second terms is 
tV D2(K-A). (20) 
Therefore, an exceedingly simple rule for 
determination of haulage can be con- 
structed. 
Before stating this rule let us make a 
further reduction in the formula. The 
unit of expressions (19), (20), is a cubic 
foot hauled a linear foot. To reduce 
this to the haulage unit, divide by 
27D.=2700. Let etc 
Yoln=-^Vn) that is, let the abbreviations 
represent the number of cubic yards 
instead of cubic feet. This, it happens, 
is the denomination used in dealing with 
these quantities, and is, therefore, the 
denomination in which these quantities 
are presented to us when we commence 
to calculate the haulage. Now, divide 
the sum of expressions (19), (20) by 27D, 
using the abreviations. 
2(H + K-A-I)+D'2(H'-H) 
+ (D—D')2(I—H') + D"2(K'—K)], (23) 
which is simply a combination of the cor- 
rection formulae for the volumes of dif- 
ferent lengths. 
The haulage from axis to fill is deter- A NEW CENTER OF GRAVITY FORMULA. 
35 
Fig. 2. 
mined in same manner. But the result 
must be multiplied by 
yards in bank, and midtiply quotient by 
number of cubic yards in cutting. The 
sum of these is the total haidage required. 
This rule, although wonderfully sim- 
ple, in view of what might be expected 
from so irregular a solid as a railroad 
cutting, is absolutely correct. It only 
remains to multiply its result by the 
price paid per unit of haulage to find the 
price to be paid for the work. 
The position of c. of g. of cut may be 
determined in same manner described 
above for the bank. 
It frequently occurs that portions of 
excessive cuttings are transported to 
spoil-banks near at hand. Often the en- 
tire top is taken off by scrapers. The 
accurate final estimate does not distin- 
guish between these portions. In such 
a case the haulage from cutting to fill, 
as found by rule, is not the true haulage. 
Let Vol' be the amount in cubic yards 
wasted, as recorded in monthly estimates. 
LetH' be the haulage of this amount to 
waste pile, as determined in monthly 
estimates. The position of c. of g. of 
this portion of cutting must be known. 
It can be found, according to method of 
last paragraph, at the time when that 
material is measured, and its position 
should be recorded. Let the distance of 
this position from the c. of g. of fill be 
L'. Now, the error in the haulage, as 
first calculated, is the result of the ope- 
ration founded on the supposition that 
YoV was moved to the fill instead of 
to the waste bank. The correction is, 
in consequence. 
Excavation Yol. 
Embankment Yol.’ 
for the reason that the material does not 
occupy same space in embankment as in 
cut. Ordinary earths become compressed 
to various degrees. Solid rock fills more 
space in the bank. The result of division 
by Embankment Yol. is the number of 
hundred feet from axis to c. of g. of 
embankment, or the true average haul 
between. This should be multiplied by 
Excavation Yol., because the amount of 
material should be measured in the exca- 
vation. The sum of the haidage from 
cut to axis and from axis to the fill is 
the total haidage from cut to fill. 
The foregoing may be condensed into 
the following systematic 
RULE. 
To find the haulage of material from 
a piece of railroad excavation to the em- 
bankment built therewith, in terms of 
the haulage unit, 1 cu. yd. hauled 100 ft. 
Consider a plane to be passed midway 
between two consecutive full stations, as 
nearly as qwssible between cut and bank. 
Multiply the number of cubic yards in 
each volume of full [100 /if.] or minor 
[less than 100 ft.~\ length, in cutting by 
the number of hundred feet its mid-sec 
tion is removed from the plane. If any 
such mid-section be on the side of plane 
toward the fill, its product must be taken 
as negative. The sum of these products 
is approximately the haulage from the 
cut to the plane. 
To correct this add the expression 
H'-L'xVol'. (24) 
Quite as often it happens that portions 
of the embankment are built of material 
from borrow-pits at hand. Let Vol" be 
the number of cubic yards borrowed, as 
measured in fill, Hx/ the haulage thereof, 
and L" the distance of c. of g. of this 
portion of fill from the plane. Let the 
number of cubic yards in entire part of 
bank to which material from cut has been 
hauled be Yol'". This is equivalent with 
expression, EmbankmentYcaused above. 
Then ('Sol'"—Yol") is the portion of bank 
brought from cut; and its c. of g.,—not 
f D2 
lff% 9x6x6 (K~A)’ 
once for every series of consecutive vol- 
umes of equal length, in the cutting, D 
being the length of each such volume, 
K, the area of end cross-section of the se- 
ries, farther from the plane, and A, the 
area of end cross-section nearer the plane. 
Determine in exactly same manner the 
haulage from the assumed plane to the 
fill/ but divide result by number of cubic 36 
A NEW CENTER OF GRAVITY FORMULA. 
the c. of g. of Yol'", as calculated in rule, 
—is the terminus of average haul dis- 
tance from cutting. If 1/" be the dis- 
tance of c. of g. of fill from plane as 
determined in rule, and Liv, the distance 
therefrom of c. of g. of {Yol'"—Yol"), 
then 
ence in labor, occasioned by change in 
amount of haulage, so often made neces- 
sary during progress of work, is directly 
provided for in the original agreement. 
The paragraphs containing expressions 
(19), (20), show that the statical moment 
of a series of consecutive, equally long 
shapes, each of which is represented by 
some of the forms of a quadratic func- 
tion, we may find by assuming the c. of g. 
of each shape to be half way between its 
ends, then compounding the several 
moments, and finally correcting by the 
expression , 
T<lv—i/"Vor"-L"Vor (25) 
Yol'"-Yol" V ' 
The correction to be made in mean haul 
distance from cutting is (Liv—L'"); and 
the correction to be made in haulage is, 
when a portion Yor of cut, has been wasted, 
(Liv—I/") {Yol-YoV) (26) 
(27) 
To determine the c. of g. of series, divide 
foregoing result by V. Therefore, find 
the c. of g. of series on supposition that 
the c. of g. of each shape is in its mid- 
section, and correct by adding to the 
distance, of the point thus found, from A 
the following: 
*D*(K-A). (20) 
When nothing has been wasted, Yol' in 
(26), (27) is zero. 
The haulage H" from borrow-pit is 
usually kept separate. 
Formulae (26), (27) are correct on the 
supposition that the correction (24) has 
already been made. 
It appears from the foregoing that in 
cases where parts of cuttings are wasted, 
or parts of embankments are borrowed, 
not only should the quantities and haul- 
age of such parts be estimated from 
monthly measurements, as is always 
done, but also the centers of gravity of 
these parts should be established and 
recorded at such times, for use in de- 
termination of total haulage finally. 
The same methods apply, of course, 
to the borrow-pits and waste-banks. The 
method can be readily modified to suit 
all practical cases. 
For another method, also depending 
upon formula (1), of determining haul- 
age, when the cuts and banks have been 
calculated entire, that is, when the con- 
tents of single volumes are unknown; 
also, for graphical methods of solving 
the problems just presented, and for de- 
tails concerning the terminal solids of 
the banks and cuts, the reader is referred 
to the chapter on Average Haul in For- 
mulae for Railroad Earthwork, Quanti- 
ties and Average Haul, since these de- 
pend partially upon formulae foreign to 
the nature of this article. 
The plan of basing contracts upon ex- 
cavation and haulage prices, seems to be 
preferred to that which considers exca- 
vation and embankment prices. The ad- 
vantage of the former is that any differ- 
(28) 
12V ' ’ 
Thus, formula (1) reduces the problem 
to this simple one: to find the resultant 
of a system of parallel forces in one 
plane, whose intensities and positions are 
given, the position of this to be corrected 
by expression (28). The singular advant- 
age of formula (1) is, that its second or 
correction-term, (28), remains as simple 
for any number of shapes in the series as 
for one. 
It is evident, in consequence, that the 
error of the assumption that the c. of g. of 
each shape is in its mid-section, is com- 
paratively less as the series is longer); 
also, that no error whatever results from 
this assumption, when the end areas are 
equal. 
For instance, to find c. of g. of a 
spherical sector whose component cone 
and segment have equal altitude, it may 
be assumed that the c. of g. of each is 
half way between its bases. 
To find c. of g. of a series of trapezoids 
such as represented in Fig. 2, assume as 
before the c. of g. of each to be midway 
its length, and correct the resulting posi- 
tion of c. of g. of series by 
D2(H + K-A-I)+D'2(H'-H) 
+ (D-DT(I-H') +D//,(K,-K). 
12 Y 
The series of trapezoids may have such 
arrangement that some are negative, as A NEW CENTER OF GRAVITY FORMULA. 
37 
illustrated in Fig. 2, p. 291, April No. of 
this Magazine. When the last ordinate 
is coincident with first, the algebraic 
sum of trapezoids is the area of a poly- 
gon, included by the top lines of the 
trapezoids, and the c. of g. or statical 
moment of this polygon can be found by 
application of formula (1) or (16). 
Suppose the polygon to be ABCDEA. 
Let a, b, etc., be the ordinates of the 
vertices; and let a', b', etc., be the cor- 
responding abscissae of the same. Since 
we may at first assume the c. of g. of 
each trapezoid to be midway its length, 
the several moments of these figures, 
when the lever arm lies in the direction 
of abscissae, are, 
Always, when choice may be had, 
place the origin at one vertex, as at A. 
Then a=o, a'=o. In consequence, one 
term in the numerator, and one in the 
denominator, of fraction, vanish, and 
some of the remaining terms become 
reduced. 
Each term of the statical moment is 
the continued product of the ordinate of 
each vertex, the difference between the 
abscissae of the two adjacent vertices, (the 
subtraction being made always in same 
direction around polygon,) and the sum 
of the abscisste of the vertex itself and 
the two adjacent vertices. 
To construct the formula for the 
ordinate of c. of g., simply change, in 
formula (33), a, b, etc., to a', br, etc., and 
a’, b', etc., to a, b, etc. 
Professor Weisbach in article 112 of 
his Theoretical Mechanics, Eckley Coxe’s 
translation, demonstrates a method of 
determining c. of g. of polygons. The 
one above presented is, however, de- 
cidedly shorter when one co-ordinate 
only of the c. of g. is required. When 
both co-ordinates are sought, there is 
little preference between them, this being 
in favor of Weisbacli’s method. 
The first two columns of the following 
example are quoted from the article in 
Theoretical Mechanics above referred to. 
The problem is solved by the method of 
this paper. 
for first, 
Ua + bW-a’Wib' + a') 
=\(a + b)(b'*-af2), (30) 
for second, 
£(& + c)(c,2-6'2), (30a) 
for last, 
£(e + a)(a'2-e'*). (306) 
The correction term, for first trapezoid, is 
.&{b'-a'y(b-a)-, (31) 
for last, 
-e’y{a- e). (31a) 
The sum of expressions (30), (31), is 
the statical moment of polygon, when 
the lever arm is parallel to abscissae. 
This may be arranged as follows: 
a' 
a 
Double Area. 
Sextuple Stat. 
Moment. 
24 
11 
x 11=121 
x 49= 5929 
7 
21 
x 40=840 
x 15=12600 
-16 
15 
x 19=285 
x-21=-5985 
-12 
- 9 
x -34=306 
x -10=-3060 
18 
-12 
x-36=432 
x 30=12960 
1984 
22444 
$la(b' - e') {e! + a’ + V) + b(c' - a') 
(a' + b1 + c') . . . + e(a' -d')(d' + ef + «')]. 
(32) 
To obtain abscissa of c. of g., divide 
(32) by area of polygon, as expressed by 
rule B, p. 293, April No. of this Maga- 
zine. Accordingly, distance c. of g. of 
polygon is 
ry. —1 23444—Q 771 
X1 — 3- T¥?4'— 1 ' 
To find yx, substitute a for a', and a' 
for a, etc.; then proceed as in above ex- 
ample. 
Applied to a triangle ABC, when the 
origin is placed at vertex A, formula (33) 
becomes 
%\_a(b' - e') [e! + a' + b') + b(c' - a') 
(ar + b’ + o') ... -f e(a' - d') (d’ + e' + a')] 
£[a(b' - e') + b(c' - a') .... + e(a' - d')~\ 
(33) 
Each term of the statical moment is 
the product of three factors, two of which 
are the factors of a term of the area. 
These two factors need be used once 
only. This makes the determination 
oasy. 
Ubc'-cb') (b' + c') _ Stat. Mom. 
£(bc'-cbr) ~~ Area. 
=i(b + c)=a?1'> /cm\ 
Likewise, J(6 + c) =y,j ' 
The abscissa and ordinate of c. of g. 38 
A NEW CENTER OF GRAVITY FORMULA. J 
of the retaining wall, Fig. 3, the origin 
being at the left end of lower base, are: 
hM(d+d'\ + (w'-d) (cl+d' + >(■'}] 
*,=6 -r (37) 
A, ,, 
-_(w + w ) 
_d d+d')+h'(w'-d){d+d'+ w')~\ 
x~ ~~l[hd' + h\w,-d)']~ 
(35) 
_\\_-dh\h + h') + d'h(h + ti) + w'hny\ 
V~ ll-dh' + d'h + w'h'] 
(36) 
According to formula (1), 
1T (w-w')h? /00v 
■V, i1'- i joy S (38) 
or, from (36), when h=h', 
2w + w' h . 
y — r — (38a) 
w + w 3 
The numerator of (36) can be reduced. 
For Fig. 4, 
Fig. 3. 
Fie. 4. 
Fig. 5. 
To determine the abscissa of the c. of 
g. of the embankment, Fig. 5, which has 
a core of different density from that of 
the covering material, we may find the 
abscissa of polygon ABCDEFGHIJA, 
and then that of polygon JIHGJ, and 
compound the moments, having regard 
to the unequal densities. But the prob- 
lem can be solved more simply in one 
operation. Suppose y to be the heavi- 
ness of covering material, and y' the 
heaviness of core material. Now con- 
sider at once the polygon ABCDEFGHI 
JIHGA, regarding the ordinates of I and 
y' y' 
H, the second time used, as — i, —h. 
y V 
To obtain the ordinate of c. of g. of 
whole dam, consider the same complete 
polygon, but regard the abscissae, when 
used second time, of all the vertices be- 
longing to the core [J, I, H, G], as if they 
were longer in ratio of y' to v- 
If it be desired to ascertain the com- 
mon c. of g. of the water resting on 
slope, and the dam itself, consider the 
polygon, AA'A"CBA, BCDEFGHIJ, JI 
HGA, and regard, when determining the 
abscissa, the ordinates between first and 
second commas as multiplied by y, and 
those after second comma, as multiplied 
by y\ When determining the ordinate 
of c. of g., multiply the corresponding 
abscissae by same quantities. 
For the determination of the statical 
moment, or the c. of g., of any series of 
trapezoids, formula (32) or (33) is pre- 
ferable to formula (16) or (1), when the 
lengths are all different. This is very 
apt to be the case in that particular ar- 
rangement of trapezoids which forms the 
polygon. (32), (33) are, therefore, emi- 
nently suitable to this shape. When- 
ever a majority of the trapezoids have 
equal lengths, as in Fig. 2, formula (1) 
is to be preferred. 
As it usually happens that the areas 
and volumes of practical shapes are cal- 
culated for other purposes in advance of 
the determination of centers of gravity, 
formula (33) is generally in simplest 
form. When the area of polygon is 
neither known nor required, expression 
(33) can be always reduced by cancela- 
tion of common factor \, and frequently 
can be much further simplified, as in case 
of triangle, and to form expressions (37), 
(38a). 
The denominator of (33), when that 
formula expresses the abscissa, is equal 
with in value, though different in form 
from, the denominator of (33) when it 
expresses the ordinate of c. of g. There- 
fore, the denominator need be calculated 
once only. Thus the denominators of 
(35), (36) are the same in value, each 
representing the area of the polygon.