DERIVATION AND ANALYSIS OF METHODS. 269 
govern every operation applied to Simpson’s rules, the same as the trape- 
zoidalrule. It is evident that even the additive method with the trapezoidal 
rule, is far simpler than the Simpson rule, and requires much less time. 
CENTER OF GRAVITY OF CURVILINEAR FIGURE AND WATER-LINE. 
1. Longitudinally.—Taking the origin at the lower left hand corner, 
the area of a vertical element of summation is expressed by ydx, and its 
moment about the origin by x«ydx. Summing for the entire area the moment, 
M =} xydx and the area, A = fyde. If x = the distance of the center 
of gravity from the origin then 
fa Mes S syde 3 (1) 
nw. af ydx 
Transpose this into the working formule, and we have 
_ szxy 
iris (2) 
In developing the values of xy in the numerator, x is equal to 0, s, 2s, 
35, 45, etc., for the various values yo, 1, yz, ys, ys, etc., of the ordinates, 
assuming the vertical axis through the origin to coincide with yo; therefore 
for any ordinate, yn, we may place x = us; and put s, the constant, outside 
the sign leaving m inside, where is the number of intervals y is distant 
from the origin. Thus 
siny _ sdny (3) 
SZY zy 3 
oS 
To find the center of gravity of a water-line, it is evident that equation 
(3) holds good, for although the moments are doubled for the two sides, 
the area is likewise doubled, and the common factor cancels out. ‘Thus 
at of xydx _2s°'iny  s&ny (4) 
This gives two equations which we will consider separately. 
2s >ny (5) 
A 
where A = area of water-line. 
