168 Analysis of Screntific Books. 
12. On the Expansion in a Series of the Attruction of a Spherord. 
By James Ivory, M.A., F.R.S. 
Mr. Ivory’s principal object in this paper appears to be the 
removal of some difficulties which occur in the demonstration 
of the method of developing the attractions of spheroids in an 
infinite series, as employed by Laplace in the Mécanique Celeste. 
«« It is natural to think,” he observes, “ that the theory of the 
figure of the planets would be placed on a firmer basis, if it 
were deduced directly from the general principles of the case, 
than when it is made to depend on a nice and somewhat un- 
certain point of analysis ;” and he conjectures, “ that the theory 
will probably be found to hinge on this proposition: that a 
spheroid, whether homogeneous or heterogeneous, cannot be in 
equilibrium by means of a rotatory motion about an axis, and 
the joint effect of the attraction of its own particles and of the 
other bodies of the system, unless its radius be a function of 
three rectangular co-ordinates ;” for ‘‘ if this proposition,” he 
continues, ‘‘ were clearly and rigorously demonstrated, the 
analysis of Laplace, in changing the ground on which it is built, 
would require little or no alteration in other respects.” 
Without, however, attempting to demonstrate this proposition 
in all its extent, the author has substituted a mode of argument 
more direct and more simple than that of Laplace, which is 
perfectly conclusive with respect to all the cases to which the 
theorem in question can possibly require to be applied. He 
has shown that by immediately transforming a given expression 
into a function of three rectangular co-ordinates, we obtain the 
same developement as is deduced in the Mécanique Céleste by 
a more general and complicated mode of reasoning, which 
seems to be so far objectionable, as it tends to introduce a 
variety of quantities into the series, which do not alter its total 
value, sinee they destroy each other, but which may possibly 
interfere with the accuracy of its application to particular cases, 
in which it may be employed as a symbolical representation ; 
for example, when any finite number of terms is assumed as 
affording an approximate value: since if the expression de- 
veloped has not been reduced to the form of a function of three 
rectangular co-ordinates, the developement may contain an in- 
finite number of terms which are introduced by the operation, 
without being essential to its final result. 
He takes for an example of such a case the equation of a 
spheroid prominent between the equator and the poles, some- 
what resembling the, figure which was once attributed to Saturn: 
and he shows that its developement in the form required will 
contain an infinite number of quantities, arising from the ex- 
pansion of a radical, which are not to be found in the original, 
function. 
