Attractions of Spheroids of every Description . 3 7 
Lagrange’s memoir. By taking the fluxion of y, making > 
only variable, we get 
1 ‘ f J _____ — (r a — r 
' V dr J [r z -zra . 
therefore I" • ~J + a 
( when r= a) 
* /* 
- o ; which must he an identical 
equation ; or such a one as, being expanded into a series of 
the powers of 7, will consist of terms that mutually destroy 
one another. Now since 
+ + “ {-£)} ■ • d °"’ 
we ought to have i s + a [%) = o : because the fluent may 
be considered as the sum of all the successive values of the 
fluxion, and an aggregate of nothings ought to be equal to 
nothing. This is the principle of Laplace’s demonstration 
stated abstractly: and it cannot be exact; for Lagrange has 
proved that it fails in a particular case, and this failure he calls 
a paradox in the integral calculus. ^ 
Lagrange has actually reduced the function f . y + a * 
dr 
into a series ; but as this would not assist us in solving the 
difficulty it needs not be noticed. In the preceding operations 
the supposition of r— a has made all the terms containing the 
factor r — a disappear, which it is nevertheless necessary to 
retain. For this purpose resume the formula set clown above, 
viz. 
(J z 
[r^—ra . 7 ) 
q ? 
-zni . y+a 1 )* 
