the Attractions of Spheroids of every Description. 23 
overturns the demonstration of Laplace; since in his proce- 
dure an infinite number of terms are neglected merely becau e 
they are multiplied by some power of the evanescent quan- 
tity $r; a reason which the preceding analysis demonstrates 
in the clearest manner to be altogether inconclusive. 
Nevertheless, if we now suppose thaty is a rational and in- 
tegral function of p/, y / 1 — p/ 2 . cos. vs', %/ 1 — p,' 2 . sin. vs', and, 
by the help of the formula (E), inquire into the values of the 
several terms in the series on the right-hand side of the equa- 
tion ( D ) , we shall find that Laplace's equation is rigorously 
true in that hypothesis. For, as we have already shewn the 
general term of the series consists of these two integrals, viz. 
— 1 
(r—f ) 1 1 • p* • y • dp 1 . d™ 
pr v—?r • r • y • ^ __ PP\ 
| r z — zrp . y-f f j LdtJL | r 2 — 2rp . y+p 1 j Lti 
which being valued separately, the result will be, 
277 277 
y — nr, -y = o: 
l — I 
therefore the right-hand side of the equation (D) will be re- 
duced to its first term, and we shall have 
dV 
297 o .7' 
— .a * 
3 
the very equation of Laplace. 
But although the proposition in the Mecanique Celeste is thus 
found to be true in one particular hypothesis, the arguments, 
that have been urged against the proof of it contained in that 
work, lose none of their force. It appears indeed that the 
quantities which Laplace has omitted are really equal to no- 
th'ng in one kind of spheroids ; yet this does not happen for 
any reason which he has assigned, but for a reason which has 
* Liv. 3e, No. 10. Equat. (2). 
