i7 
the Attractions of Spheroids of every Description. 
matter spread over the surface of the sphere.* From this 
circumstance indeed is derived one great advantage of his 
method, namely its great generality; for no restriction what- 
ever is imposed on the nature of the spheroid excepting that 
of a near approach to the spherical figure. Nevertheless the 
author, by means of a simple transformation, immediately 
deduces from his theorem an equation which proves that y 
and V are expressed by two series both containing the same 
sort of terms :-f and since all the terms of the series for V 
can only be rational and integral functions of p, s / i — f . 
cos. -2T, V 1 — f . sin. -sr ; J it follows that y must be a like 
function of the same three quantities. We may remark here 
that this consequence of Laplace’s reasoning appears to be 
inconsistent with the premises : for it is hard to reconcile with 
the rules of legitimate deduction that an equation obtained by 
supposing y to be arbitrary, should, merely by having its form 
changed, be made to prove that the same quantity must be 
restricted to signify a function of a particular kind. But we 
mention this only by the bye, without meaning to insist upon 
it ; although we cannot help thinking that it ought to have led 
the learned author to entertain suspicions of the accuracy of 
his calculations ; all that we intend by the foregoing observa- 
tion is to prove that in point of fact we shall embrace the 
whole extent of Laplace’s method by supposing y to be a 
rational and integral function of three rectangular co-ordinates 
of a point in the surface of a sphere. 
Supposing then v to denote such a function as has been 
mentioned, we are to investigate the value of this integral, 
* Liv. 3e, No. io. 
MDCCCXII. 
t Liv. 3c, No. 11. 
D 
% Liv. 3e, No. 9. 
