Attraction of Spheroids differing little from a Sphere. 387 
U® = sy at® Yo. 
substituting this in the general expression for V’, 
V = eee + aor é. hae + - Y + _ YO &e.} 
In those parts of this investigation which are here set down, 
I think it is plain that’ there is no material error. The only 
thing which is taken for granted, is, that the expression for y can 
be resolved into a series of the form Y” + VY” + &e. in one manner 
only. In the proof of this important proposition which Laplace 
has offered, and which Mr. Ivory has mentioned several times 
without making any objection to it, there is, I apprehend, a radical 
defect. In fact, Laplace has made with the utmost confidence 
a statement, for which there does not appear to me to be the 
slightest evidence. I will now describe the nature of Laplace’s 
demonstration, and point out the parts which seem to be de- 
fective. 
Laplace first shews, in the most satisfactory manner, that 
the double integral of Y° .Z® with respect to » and o, (the limits 
of « being — 1 and + 1, and those of w being 0 and 27,) is 0 except 
k=t The remainder of his demonstration is in substance as 
follows. If possible, let y be resolved into the series Y" + Y" + &e. 
and also into the series Y'” + Y" +&ec. Multiply y by Z° and 
integrate between the limits mentioned above: then, in consequence 
of the proposition already demonstrated, the integral in one case 
will be reduced to the integral of Y".Z", and in the other case 
to the integral of Y"’.Z°. If then y can be resolved into two 
such series, we shall have the integral of Y. Z equal to the integral 
of Y’’.Z", or the integral of {VY — ¥"’} Z°=o0. But, says Laplace, 
it is easily seen that if we take for Z”, the most general function 
of its kind, this equation cannot be true except VY — ¥ =o, 
Vol. W. Part Il. 3D 
