75] 
ON THE ATTRACTION OF AN ELLIPSOID. 
437 
and it is obvious that for this value of co we have A = 8. The expression for 
the attraction is easily reduced to a known form by writing y = this gives 
At (k + lyf (k + my) 1 (k + nyf wda> 
A — 4>irla 
h 
Pa 3 (k + my) (k + nyf + m 3 b 3 (k + nyf (k + lyf + n 3 c 2 (k + lyf (k + myf' 
Also 
> 3 = k 
Pa 3 
+ 
m‘ 2 b~ 
+ 
nx- 
k + ly k + my k + ny) ’ 
whence 
^ _ _ k [Pa 3 (k + myf (k + nyf + m 3 b- (k + nyf (k + lyf + n 3 c 3 (k + lyf (k + myf ] 
2 (k + lyf (k + myf (k + nyf 
and thus 
A — ‘¿'irkrla 
dy 
{k + lyf (k + my) 2 (k + nyf 
where for the entire ellipsoid the integral is to be taken from y — ^ to y = oo. A 
k + lP 
better known form is readily obtained by writing x 2 = > i n which case the limits 
for the entire ellipsoid are x = 0, x — 1. 
It may be as well to indicate the first step of the reduction of the integral I, 
viz. the method of resolving the denominator into two factors. We have identically, 
(A — 8) (IP 2 + mQ 2 + nB?) = (o 2 (P 2 + Q 3 + R 3 ) + A (IP 3 + mQ 3 + nR 2 ) — (laP +- mbQ + ncRf, 
and the second side of this equation is resolvable into two factors independently of 
the particular values of P, Q, R. Representing this second side for a moment in the 
notation of a general quadratic function, or under the form 
AP 3 + BQ 3 + CR 3 + 2 FQR + 2 GRP + 2 HPQ, 
we have the required solution, 
IP 3 + mQ 3 + nR 3 = 
2 [AP + {H+V(- €)}«+{(? + v(- 33)} -K] [AP +{H- V(- ®)) Q + {<? - V(- 33)) fi]; 
where, as usual, 33 = GA — G 3 , = AB — H 3 , and the roots must be so taken that 
V(- 33) V(- «) = iF (iF = m-- AF)). 
I have purposely restricted myself so far to the problem considered by Legendre: 
the general transformation, of which the preceding is a particular case, and also a 
simpler mode of effecting the integration, are given in the next part of this paper.