Attractions of Spheroids of every Description . 
S9 
the limits -f 1 an ^ — l, 
/ — (r — a) ■ c . dx f 1 r—a ■) 
(r— axY ~ 2 a ' \r—a (j-VaY}' 
And in this instance, the fluent is infinitely great when 
r = a. 
All the three fluents which we have just been considering, 
ought to be alike equal to nothing, according to the reasoning 
of Laplace. For, when r — a, all the fluxions are evanescent 
for every value of x between the proposed limits, with the 
exception of the single case x = 1 in the two last, for which 
value of x the fluxions are infinitely great. And even in the 
m. 
first instance if we change the factor r — a into (r — a) 7 , 
making ^ less than then in this case also the fluxion will 
be infinitely great when x— i, while the whole fluent will 
still be evanescent as before. If therefore we would have an 
unerring criterion to direct us in such instances, we must con- 
sider the expression of the fluent. If that expression is finite 
at both the limits, and likewise for every intermediate value 
of the flowing quantity, then, on account of the evanescent 
factor, the whole integral will be equal to nothing : but if that 
expression becomes infinitely great at either of the limits, or 
for any intermediate value of the flowing quantity, then the 
whole fluent will be equal to a finite quantity when the eva- 
nescent factor is raised to the same power in the numerator 
and denominator ; and it will be infinitely great, when the 
evanescent factor is raised to a higher power in the denomi- 
nator than in the numerator. The examples we have given 
above fall under these three cases, and they are quite ana- 
logous to the distinction of cases in Laplace’s theorem, as 
noticed in No. 6 of my paper. We may add farther that the 
