442 
ON THE ATTRACTION OF AN ELLIPSOID. 
[75 
The remainder of the process of integration may in many cases be effected by the 
method made use of by Jacobi in the memoir “De binis quibuslibet functionibus &c.” 
Crelle, t. xii. [1834] p. 1, viz. the coefficients a, a', &c., /3, &c., may in addition to the 
conditions which they are already supposed to satisfy, be so determined as to reduce 
any homogeneous function of p, q, r, ... entering into the integral to a form containing 
the squares only of the variables. This method is applied in the memoir in question 
to the integrals of n variables, analogous to those which give the attraction of an 
ellipsoid; and that directly without effecting an integration with respect to the radius 
vector. I proceed to show how the preceding investigations lead to Legendre’s integral, 
and how the method in question effects with the utmost simplicity the integration 
which Legendre accomplished by means of what Poisson has spoken of as inextricable 
calculations. 
Consider in particular the formula 
(x, y ...) h dxdy... 
(a? + y 2 .. .)%~ l 
the number of variables being as before n, and (x, y, ...) h denoting a homogeneous 
function of the order h. The equation for the limits is assumed to be 
(where 8, = la 2 + mb 2 ... —k, is taken to be positive); or more simply, 
yjr (x, y ...) = (a) 2 + 18— l 2 a 2 ) x 2 + (&) 2 + m8 — m?b 2 ) y 2 + ... — 2 Im ab xy — ... 
Here fx = h + 2i — 3. Also, putting for shortness 
lap + mbq + ... = A, Ip 2 + mq 2 + ... = d>, 
it is easy to obtain 
[(A + wp) h+2i+n ~ 3 - (A - wp) h+2i+n ~ s ] 
Also 
values which give 
h -p 2 % + n — 3 
r (_)*<«—a a p*i-1 [(A + œp) h+2i - 
J- «i 
t+n-3 (A - 6)^+8*+«-»] (p,q,...) h Sdco dd ...