516 
ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’s METHOD). 
[89 
tial of a spherical surface radius unity, and if 9, <f> are the parameters by which the 
position of p is determined, we have 
dS = 
In the present case 
dS = dS, = 
X, 
Y, 
Z 
dX 
dY 
dZ 
dd’ 
dd’ 
dO 
dX 
dY 
dZ 
d(f> ’ 
dcj) 
deb 
x lt 
Fi, 
z, 
dX 1 
dY, 
dZ, 
dfr 
dY,’ 
df, 
dX x 
dY, 
dZ, 
d Z, ’ 
dZ, 
dZ, 
d9 d(f). 
dY,dZ,, 
or from the values of X„ F 1( Z 1 in terms of X„ Y„ Z, (observing that X, must be 
replaced by its value \/(l — Fi 2 — Z, 2 )j we deduce 
dS = 
(u ;)(« r)| 1 d7iz 
(u-q)(u-r)) X 1 
But dtS = dS, — i- dY,dZ„ 
X, 
whence 
dS = 
(u — q) (u — r) 
(u -q) (u- r) 
X.dS 
X, 
which shows that the corresponding elements of the spheres whose centres are P, P, 
projected upon the tangent planes at P and P respectively, are in a constant ratio. 
It may be noticed also that if pu, p, are the masses of the ellipsoids, the ratio in 
question 
(a — q)(u — r) 
(u — q) (u — r) 
fXCli 
pci) 
We have thus 
that is 
A 
(u — q)(u — r)| ipXjXdS 
(u — q)(u — r)} J X 1 
A = 
q) (u - r)) 
q) (u - r)\ 
fp(aX 1 + a , Y 1 + a"Z 1 )X 1 dS 
J x; 
The value which it will be convenient to use for p is that derived from the equation 
P 2 
( X 2 F 2 Z 1 \ 
\/+ u + g + u + h + u) + P 
X a Yb Z'c \ u — u 
1 [ i i 
,/+ u g + u h + u) aA