Attraction of Spheroids differing little from a Sphere. 389 
ie pe 1=N) (Ve 1 th ee ) } 
=|" {ni pe we) fig 2 + &e.| 3 { An) sim zw + Bo cos nwt, 
upon giving to ~ the values 0, 1, 2, &c. Now since the first term 
of the series multiplying sin x», or cos nw is always p'~". (1-1), 
it is evident that no combination, by addition, or subtraction, 
of the series, multiplying cos nw in the expressions for Y,. ¥, 
¥®,.....¥°~” can be equal to that in the expression for F”. If 
then we resolve y into a number of such terms as Y, F, VY, &e. 
by putting it into the form 
P+Q cos o+R cos 20+ &e. +7 cos sw ) 
+ QSnot+ rsin20+&ce.+ ¢tsin sw f’ 
and first determme Y, so that, upon subtracting it from this 
expression for y, the terms cos sw and sin sw, and the first term 
of each of the series multiplying the cosines of the other multiples 
of », may be taken away; then determine F“-” in the same 
manuer; and so on to Y"; as Y cannot be expressed by any 
multiples of ¥°~”, Y°~, &e. the constants entering into its ex- 
pression will be determinate, and will admit of only one value. 
The same reasoning then applies to the constants entering into 
the expression for F“~”, Y“~, &e. and thus it appears that if y 
be a rational function of «4, ./1—»7 cos w, and ./1—x' sin o, it ean 
always be resolved into a series of the form V'+¥V+&ce. and 
can be so resolved only in one manner. 
Since the theory of the third book of the Mecanique Celeste 
depends entirely on this proposition, I conclude with Mr. Ivory, 
that that theory applies only to spheroids in which the elevation 
of the spheroid above the sphere is expressed by a rational function 
of », /1- cosw, and ,/1— sinw. If any other function can 
be expanded into a converging series of this form, it is plain that 
it must be included in this statement. 
I shall conclude this paper by briefly mentioning the order 
3D 2 
