contained in the Third Book of the Mécanique Céleste. 38 
of it. If we suppose that the thickness of the stratum is a 
finite and integral function of three rectangular co-ordinates, 
we embrace all the applications to the figure of the planets; 
the demonstrations are clear and effected by the usual pro- 
cesses; and near the point of contact the thickness is divisible 
by the distance from that point, which is much more general 
than the cases comprehended in the demonstration of Laplace. 
By substituting the value of V expanded in a series, in the 
equation that takes place at the surface of the spheroid, the 
author of the Mécanique Céleste proves that the function y in 
the value of the radius may always be expressed in a series of 
terms, each of which is determined by an equation in partial 
fluxions, to which it is subject. This is a fundamental point in 
the analytical theory; and as it is a consequence of the equa- 
tion at the surface, it can be considered as true only in the 
cases in which that equation is clearly proved. Yet in the 
whole course of the work the symbol y is considered as per- 
fectly general, and as standing for any function; which can- 
not fail to embarrass the reader, since the proof of the equa- 
tion is deficient and limited. Instead of deducing the develop- 
ment in question from the equation at the surface, it will be 
much more simple to deduce it from the same formula on which 
the equation itself has been shown to depend: by this means 
the whole theory will rest upon a single analytical proposition. 
Now, resume the second of the formule (2), 
J (P-@) x wee =i) 
and separate it into the two parts of which it consists, then, 
r2—a?) ds r—a2)y'ds 
y x ( “ ay a yas 
As f is a function of ), if we put ds = a? dy sind 4, it be- 
comes easy to find the integral on the left side of the equa- 
tion. For the arcs ) and ¢ are independent on one another ; 
and the integration being effected, first with regard to d¢, be- 
tween the limits ¢ = 0 and ¢ = Zz; and then with regard to 
dw, between the limits { = 0 and = 7, the result will be 
equal to 42a when r=a. The last equation will therefore 
become 
(r2—a?) yds | 
ee ; 
4nay= 
and if we consider z/ as a function of the arcs 4 and a!, we 
shall have ds = a’ d# sin §'d a, and 
uipad (72—a®) ary! dé sin & da" 
hie 4ra f3 S (3) 
Vol. 67. No. 333. Jan. 1826. E Again, 
