Ill 
in a series of the attraction of a spheroid. 
co-ordinates, the formula (A) cannot be integrated. And, 
perhaps, what is now said, is alone sufficient to show that, in 
this case, some modification takes place, which it were de- 
sirable to have fully explained. On attempting to transform 
y into an expression containing <y, Vi — y 2 , sin 4/, cos 4/ in 
place of cos O', sin 6', sin <p' , cos <p', the powers of Vi —y 
make their appearance as divisors ; and hence it is to be feared 
that the integral will be infinite at the limits ; which circum- 
stance would make it impossible to conclude with certainty 
what the value sought will become in the particular case of 
a = r. But it would be of no utility to seek a strict demon- 
stration of the differential equation : because in reality the 
method, when it is extended to all functions of tw'o variable 
arcs, is independent of that equation, being derived from this 
proposition, that every such expression is either explicitly a 
function of three rectangular co-ordinates, or may be trans- 
formed into one. The developement in question may always 
be found, as has been showm, by the rules of algebra ; and 
the differential equation is w'anted neither for proving the 
possibility of the developement, nor for calculating its terms. 
But in this plainer way of considering the matter, it appears 
that the developement does not represent the given expression 
y, when that expression is not an explicit function of three 
rectangular co-ordinates, in the same sense that it does when 
it is such a function. There is, therefore, a difficulty left 
unexplained ; and we may be permitted to doubt, whether so 
important a part of the celestial mechanics, as that regarding 
the figure of the planets, rests, with sufficient evidence, on 
the doctrine laid down concerning the generality of the 
developement. 
