89] ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’s METHOD). 517 
with only the transformation of expressing the radical in terms of X,, viz. 
Xa Yb Zc 1 v 
— 1— _j_ 1— X, 
_j + u g + u h + u a, 
P ~ ^ X 2 ~ r- Z 2 
- f 1 
f+u g+u h+u 
substituting these values and observing that F, and Z x are rational functions of 
X, Y, and Z, but that X 1 is a radical, and that in order to extend the integration 
to the whole ellipsoid, the values corresponding to the opposite signs of X 1 will require 
to be added, the quantity to be integrated (omitting for the moment the exterior 
constant factor) is 
f / Xa Yb_ Zc \ 
\f + u^ g + u^ h + u) 
+ 1 -(a:Y 1 + a"Z 1 )\x i dS 
+J 
X 2 Y 2 Z 2 
/ + u g+u h+u 
the integration to be extended over the spherical area S. Consider the quantity within 
{ }, this is 
/ Xa Yb Zc 
+ —— +7- 
a 
+ - 
q u ^( a 'X + (3 r Y+fZ)+~ 
\f+u g + u h + u) a-i\ \q — u 
The coefficients of Y and Z vanish, in fact that of Y is 
r — u 
r — u. 
(oTX + /3" Y + y"Z). 
r — u\ cf 
f+u g + u 1 a 1 {f+q)\/ \q — u) g + q ' a l {f+r)\f \r — u) g+ r 
m b \a 
+ 
q — u^ bf c,a 
ab 
+ 
b 
q — u 
ai K/+ u) (g + u) (/+ q) (g + q) V \q ~ W (/+ r) (g + r)\f \r - u, 
+ 
C+! 
ab x 
a+ 
+ 
w 
+ 
«x 1 (f+u)(g + u) (f+q)(g + q) (f+r){g + r) 
and similarly for the coefficient of Z. 
= 0; 
The coefficient of X is in like manner shown to be 
aa f a * b ' Cx 2 ) _ cm (f-g) (f-h) 
Oil(/+^) 2 {f+qf (.f+r) 2 } a x {f+u){f+q){f+r) 
aa 
a 2 a. 
a 
aa, : 
a 
aa. 
X. 
or the quantity in question is simply