146 Mr. Ivory on the figure requisite to maintain the equilibrium 
The figures of the earth and of the planets being entirely 
deduced from the properties of spheroids little different from 
spheres, it may not be improper to conclude this Paper with 
a short exposition of a theory that occupies so conspicuous a 
place in the celestial mechanics, and which is so intimately 
connected with the subject we have been discussing. 
For this purpose resume the expansion of V (r) already 
given in § 3, viz. 
V (r) =/JR”dv!dv'+ rJJK. C (,> d/dv' 
+ r *. j-^+//Log.R'.C w ^'rf!x'} 
r 1 ffC^dy!dv 
f ^ d (a! dzs' 
R'«- 2 
&C. 
The spheroid being nearly a sphere, we may suppose 
R'ss=a.(i + *.y); cc being a small coefficient of which the 
square and other powers are to be neglected ; and y' a func- 
tion of the angles that determine the position of R'. The 
expansion supposes that the attracted point is within the 
spheroid ; but it will apply when the same point is in the 
surface, in which case, r = R = # ( 1 -f 01 -y)- Now, let the 
values of R and R' be substituted, and we shall obtain, 
V(R) = jj~ (i + 2 a/) dp! dvr' —a 2 (1 + 2 ay) 
-f - a 2 cc .JJy' • d\d d’ur 
+ a* ujfy'. C^dn'd'*' 
-f- a 2 a .ff y . d\fi dvr 
+ &c. 
