75] 
ON THE ATTRACTION OF AN ELLIPSOID. 
443 
where it will be remembered that p, q,... are linear functions (with constant terms) 
of (n — 1) variables y, these last mentioned quantities being themselves functions 
of (n — 2) variables 0, &c. such that 1 — y 2 — £ 2 — ... = 0 identically. If besides we suppose 
1 'ff' 
that <E> = Ip 2 + mq 2 + ... reduces itself to the form p~ q~ &c., we have, by the formula 
of the paper “ On the Simultaneous Transformation of two Homogeneous Equations of 
the Second Order,” [74], 
(to 2 + 18 — IX) ((ù 2 + m8 — mX) ... f 1 
which is true, whatever be the value of X. 
Pa 2 
m 2 b 2 
It seems difficult to proceed further with the general formula, and I shall suppose 
71 = 3, i = 0, h = 1, (x, y...) h =x, or write 
y f xdxdy dz 
J (nr? 4- nfl 4- 
(x 2 +y~ + z 2 ) 
the equation of the limits being 
l(x — a) 2 + m(y — b) 2 + n (z — cf = k. 
Here we may assume y = cos 6, £ = sin 0, (values which give S = 1). And we have 
V=2 
aPd(o f (a + a! cos 0 + a" sin 0) d0 
K* J 1 cos 2 0 sin 2 0 
p-- 
Q R 
from 0 = 0 to 0 = 27t ; or, what comes to the same thing. 
aPdœ i ocd0 
si 
V=8 
from 0 — 0 to 0 = ¿7T. Hence 
V = 477 
] 1 cos 2 0 sin 2 0 ’ 
p Q~ 
ouo 2 P\/(QR) do) 
\/{{P - Q) (P-R)} ’ 
we have from the formulae of the paper before quoted, 
„ QR (B- mP) (G-nP) - F 2 
K (P-Q)(P-B) ' 
B, C, F, being the coefficients of y 2 , z 2 , yz in yfr (x, y, z), viz. 
B = (o 2 + m8 — m 2 b 2 , G = a» 2 + n8 — n 2 c 2 , F = mnbc ; 
and consequently 
V — 477 I co 2 da> 
PQR \(B- mP) (G - nP) - F 2 }* 
(P — Q)(P — R) 
56—2