CEETAIN APPLICATIONS TO THE THEOET OE DEFINITE INTEOEALS. 787 
Po, Pi, ..Qo, Qi-. being constants whose values in terms of pi, pg, ..qi, qg.. can always he 
finitely determined. 
As particular examples of the above, we should have 
f{v)dv 
f” ^ — f” 
— V ^“"^1 ^ — \ 
C" ^ ^ an \ __ f 1 , I an ^ r f{v)dv _ 
x-h^J “r(4-A„)2j\ / ^ «i ^ y ‘ 
h-xf h-kj 
In the last theorem the particular case in which h — Ki, the function /('y) being at the 
same time supposed small for very large values of v, is interesting. We see that as h 
approaches Ki the terms p and become large in comparison with those with 
which they are connected by addition. Thus the second member approaches the value 
so that in the limit 
«i fiv)dv 1 f” , 
P” dx / 
x — xj 
It may be worth while to verify this theorem in a particular instance. We have from it 
Now assume in the fii’st member 
— a 
The transformed integral is easily found to be 
As to the limits, when x varies from — oo to X, ^ varies from X to oo ; and when x varies 
from X to CO , y varies from — oo to X. Thus, by mere transposition of the two portions 
of the integral, the limits of y become the same as those of x, and we have 
As a particular deduction from the above we shall have. 
.c 
dx Tflg 2a 
g ^y= , 
which may also be veritied by differentiating (5.), art. 32, with respect to a. We are 
permitted thus to differentiate with respect to a, because the function under the sign of 
integration does not become infinite within the limits. This condition must be strictly 
attended to in all similar attempts at verification. 
5 K 2 
