109 
in a series of the attraction of a spheroid. 
y= i ; which can happen only when 9' =9, and <p'=<p. If 
therefore we suppose r* — a 2 to be indefinitely small, the 
little from 9 and <p. But when 9' and differ indefinitely little 
from 9 and <p,y' or f(9\ <p'), will differ indefinitely little from 
y, ovf ( 9, <p) : and hence it follows that, in the whole extent 
of the integral, we may consider y' as constant and equal toy. 
We have therefore to prove the truth of this formula, viz. 
between the limits 9'=o , <p‘=o, and 9’= tt, $>'=27t, in the par- 
ticular case of a=r. 
The two arcs 9, 9' and the arc of which y is the cosine, are 
the three sides of a spherical triangle ; <p — is the angle op- 
posite to the last arc ; and if -vf/ — ;]/ denote the angle oppo- 
site to 9', the element of the spherical surface will be equally 
expressed by sin 9' d9‘ dq> , or d<fd. cos 9, and by dtydy. where- 
fore we have to integrate this formula, viz. 
then take this integral between the limits y = - f- l, and 
7 = — l ; and 
whole value of the integral will be obtained by extending the 
integration to such values of 9' and q>' as differ indefinitely 
Now integrate between the limits S/=o and \f/= 27 t : then 
integrate again, and 
V r % — a 1 
