CEETAIN APPLICATIONS TO THE THEOET OF DEFINITE INTEGEALS. 779 
This is the general theorem of definite integration to which reference has been made. 
The remainder of this paper will be devoted to its illustration. 
The theorem is independent, as has been said, of the nature of the functional inter- 
pretation off, and, even when the factor /’(.t — ^ ) becomes infinite for the 
limiting values X,, does not fail, but carries with it a correction for the discon- 
tinuity thence arising. We cannot otherwise attach a meaning to the expression 
^ f{x)dx, when for a value x—\, included within the limits of integration, becomes 
infinite, than by considering it as the limit to which the sum of the integrals 
f y{x)dx-{-^ f{x)dx 
tend as e and d tend to 0. According to the nature of the function and the modes 
in which e and d tend to the limit 0, the integral may become, as Cauchy has observed, 
finite or infinite, determinate or indeterminate. When d=e, so that the approach on 
either side to the limiting value X is made in the same manner, ^. e. by equivalent infini- 
tesimal variations of x, the value of the integral obtained will be that which Cauchy 
terms it?, principal value. The equation (1.) will thus give the principal value of the 
integral in its first member, if we suppose v to approach by the same kind of variations 
to the limits — oo and oo ; in other w'ords, if, representing the function under the sign of 
d* 00 
integration in the second member by F(^;), we regard I Y[v)dv as the limit of the value 
c/ — 00 
of j Y[v)dv, a becommg indefinitely great. For suppose x to be approaching the par- 
^ —a 
ticular limit Aj. The nearer its approach the more nearly {vide 6, art. 30) is the follow- 
ing equation realized, wz. — 
—a. 
whence the more nearly have we 
-=y. 
. Ml - 
— 1- 5 
1 I y 
and therefore, if v tend towards oo and — oo by equivalent variations, so also \vill x by 
equivalent variations approach from above and from below the limit 
Again, the larger x becomes the more nearly do x and v approach a ratio of equality, 
and therefore the mode of approach of x in the first member to the limits — oo and oo 
determines identically the mode of approach of to — oo and oo . Thus we may finally 
give to the theorem the following rigorous statement, viz. — 
TJie two members of the equation 
1 
^ — Xj** OO^Xn 
approach a ratio of equality as a approaches to infinity, provided that if 
^ — Xj OC — Xn/ 
5 I 2 
