514 
ON THE ATTRACTION OF ELLIPSOIDS (jACOBl’s METHOD). 
[89 
and the corresponding ones 
a 2 b~ c 2 u — u 
f +u g + u h + u a 2 
X a Yb Z c u — u / Xja x FA Z& \ 
f+u g + u h + u a* \u — u u — q u — r) ' 
t X a Yb Zc y ^u — u / X 2 F 2 Z 2 \ 
\f+u g + u h + u) ' ad \f+u g + u h + u) 
_ u — u / Xd + Fd + AM _ Xd 
ad \a — u u — q u — r) ad 
The coordinates of the point Q are obviously a + pX, b + pY, c + pZ (where p = PQ); 
substituting these values in the equation of the interior ellipsoid, we obtain 
/X 2 F 2 Z 2 \ ( Xa Yb Zc \ ( a 2 b 2 c 2 \ _ 
^ V f + u+g + u^ h + u) f+ u + g + u + h + u) + \ f + u g + u + h + u ) 
reducing the coefficients of this equation by the formulae first given, and omitting a 
factor M , we obtain 
«1 
f odX 2 / XjCh YJh Zfi, 
\(u — u) 2 \u-u + u 1 -q^u-r. 
+ — 
p'-+2p(+^+ Y -^-+ 
' ' \u — u u — q u — rj 
that is, 
adX] 
(ü — u) 
2 P' 
a. 
+ 
F161 , A<á 
+ 
P = 
.u —u u — q u — r. 
1 
x+h Xjttj ^ Y+Lx ^ ZxCx 
ü — u ü — q u — r 
-1 , 
which is easily transformed into 
P 
u — u 
iX] — OjXj + (/ ^1 A (,/+ ' u ) ^1 ^ i (J + u) CxZx — ( f + u) CxZx 
f+q f+r 
and this form remaining unaltered when u and u are interchanged, it follows that if 
PQ — P) then p — p, which is a known theorem. The value of p or p may however be 
expressed in a yet simpler form ; for, considering the expression 
X 
X 
flfiXi b] 11 CxZx 
+ r 
_ ^ f &iXx CxZx 
V(/+w) V(/+ u) \J(f + ü) \f +U~ r f +q~ r f+ rj V(/+ u) {/+ii f+q J+r 
- 1 
u - u W( f+ u) V(/+ u) 
dxXx — OjX, | (/ + (/+ ' u ) A (./ + ujcXi — (/ + uRA 
/+g /+r