### On the Expansion in a Series of the Attraction of a Spheroid

J. Ivory
1822 Philosophical Transactions of the Royal Society of London
C 99 3 XII. On the expansion in a series of the attraction of a Spheroid. B y J ames Ivory, M . A. F. R . S. Read January 17, 1822. T he 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. T he whole o f this doctrine depends on one fundamental proposition. Let f ( 9, q>) denote any function of the sines and cosines of the variable arcs 9 and <p) = Q° + Q (,) + Q (2) • • •
more » ... Q (0 • • • &c. every term o f which will separately satisfy this equation in partial fluxions, viz. 1-t** Now in one case there is no difficulty. W h e n e v e r /(0, <p) stands for a rational and integral function of [x, V'i -. sin <p, ■^l-*[x2. cos <p; 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 f( 9, <p)is not such a function as has been men-on July 19, 2018 http://rstl.royalsocietypublishing.org/ Downloaded from the symbols M^, M^, &c. standing for rational and integral functions of cos 0 and sin 9, or of [x and V i -jx2. Again, every even power of Vi -(x3 is an integral function of ^; and every odd power is equal to a similar function multiplied by V i -: the value of y will therefore be thus expressed,