36 Mr. Ivory on the Theory of the Figure of the Planets. 
ment of the quantities of which they consist. The equations 
in partial fluxions to which the terms on the right-hand side 
are subject, are derived from the expressions C‘!), C(®, &e.3 
and they are such as to determine each term separately when 
the aggregate of the whole is given. 
I have now examined particularly the fundamental points 
of the analysis of Laplace. Such an examination was required 
in a theory which in other respects is not unexceptionable. 
In intricate cases, in which there occur difficulties of different 
kinds, it seems best to acquire correct notions on one part 
before we proceed to the other parts. If such discussions (but 
little calculated to make a brilliant display in the eyes of the 
public) be ill suited to the prevailing taste of the present times, 
it must be acknowledged that they are necessary, unless we 
would entirely neglect a branch of knowledge that has always 
been reckoned of great value. 
But it would be improper to pass on to the second branch 
of my subject without noticing a demonstration of the equation 
at the surface of the spheroid, which we owe to M. Poisson*. 
This celebrated mathematician, who has particularly studied 
this branch of analysis, considers the formulee (3) and (4); 
and he proposes to demonstrate their truth, supposing that 
y stands for any function whatever of the two arcs 4 and o. 
We are not therefore left in any uncertainty about the extent 
of the proposition to be proved. He observes, that on account 
of the evanescent factor the element of the integral is equal to 
zero, in all positions of the molecule, except when it is ins 
finitely near the point of contact of the two surfaces, when the 
denominator is infinitely small. Now, at the point of contact, _ 
we have 7/= y, '= 6, a’! = @; wherefore, if we put #= 6 +f, 
o'=a + k, we shall obtain the value of the double fluent by 
extending the integration to infinitely small values, positive or 
negative, of h and &. But while the arcs 4 and @ acquire the 
infinitely small variations / and /, the thickness of the mole- 
cule z/ may be supposed to remain constant; or, which is the 
same thing, we may put the equation (3) in this form, viz. 
LPS (7? —a?) ay’ dé sin&g da 
‘bed 2 aif" eaen Toe 
He then finds the value of the integral in the manner he pro- 
posed; but as the same value may likewise be found by the 
ordinary rules, this part of the process adds nothing to the 
main argument. The force of the demonstration turns en- 
tirely on the assertion, that we may integrate on the supposi- 
tion that the thickness of the molecule remains constant. 
* Journal de? Ecole Polyt, tom. xii. p. 145. 
To 
