XII. On the expansion in a series of the attraction of a Spheroid. 
By James Ivory, M. A. F.R. S. 
Read January 17, 1822. 
The purpose of this paper is to make some observations on 
the developement of the attractions of spheroids, and on the 
differential equation that takes place at their surface. 
1. The whole of this doctrine depends on one fundamental 
proposition. Let f( 9 , <p) denote any function of the sines and 
cosines of the variable arcs 9 and <p ; and put p, = cos 9 : then 
the given function may be developed in a series, viz. 
/(«. 9) = Q° + Q (,) + Q (2) • • • + Q (i) • • • &c. 
every term of which will separately satisfy this equation in 
partial fluxions, viz. 
.(0 
d[j. 
d<p 
Now in one case there is no difficulty. Whenever/(0, <p) 
stands for a rational and integral function of fx, Vi — ^ . sin <p, 
V 1— p,“. cos $ ; or of three rectangular co-ordinates of a point 
in the surface of a sphere; the proposition is clear. In this 
case the same combinations of the variable quantities are 
found in the terms of the series and in the given function ; 
and by employing the method of indeterminate coefficients, 
the two expressions may be made to coincide. The inquiry 
is therefore reduced to examine the nature of the develope- 
ment when/^ ( 9 , <p) is not such a function as has been men- 
