extensive Class of Spheroids. 71 
V= - i=-* + a* .«*+ sw . «a*. { U (o) + £ . U (l) + £. U (3) 
+ + & c - } • ( 6 ) 
The formulas (5) and (6) enable us to compute the attrac- 
tions of homogeneous spheroids on a point without or within 
the surface ; and, for a point in the surface, we may make use 
of either series, observing to put r = a in all the terms mul- 
tiplied by «, and r = a . a .y) in the rest. Wheny' is a 
finite function, the two expressions for V will both stop. It 
would be easy to deduce from hence the attractions of hete- 
rogeneous spheroids; but having nothing new to offer on this 
head, I shall refer the reader to Laplace’s work. No. 14, 
Chap. 2, Liv. 3e. 
The two serieses marked (5) and (6) will be found to be 
entirely equivalent to the formulas (3)* and (4)”)* which La- 
place has given in the second chapter of the third book of 
the Mecanique Celeste ; for in effect the coefficient of & . 
J + 3 
r 1 * 1 ’ 
yl 
in two of the serieses ; and the coefficient of a . - — ■, in the 
a l—2 
other two, are only different expressions of the same integral 
y . . dts' . Qf l \ the symbol y' being always understood 
to denote a rational and integral function of three rectangular 
co-ordinates of a point in the surface of a sphere. In point of 
result therefore the two methods are one and the same, and 
the solutions they furnish are both applicable in the same cir- 
cumstances. Neither of them can be of use, unless the radius 
of the spheroid be first reduced into such a function asy is 
supposed to denote. The one solution can claim no preference 
* No. 1 1 . 
f No. 13. 
