AND ON DEFINITE INTEGRATION. 479 
This theorem, which will be found to be of 
extensive utility in the definite integration of 
those functions which can be integrated only by 
means of infinite series, is remarkable, from the 
circumstance of its expressing the value of the 
x 
definite integral .f (ex) dx by means of two 
Jf 
distinct quantities which are, first, the summation 
of a definite number of terms of the series 
whose general term is f jo(a+h)t, and secondly, 
the summation of an infinite series whose general 
n—I n—I 
termis 5 (h'))f (ow)—f (ay)}, which sum- 
mation we shall now designate the complement of 
a 
the definite integral | »f (ae) da. 
y 
Equation (13) will enable us to determine ¥(«) 
when F f (oxv)dx and & ; fjo(w-+h) tare known, 
JY g 
and consequently, when any two of the above 
distinct quantities are given, we can find the 
third. 
Cor. 1. If we take *(v)=«#, we shall have 
from equation (13) 
