162 
MR. W. H. L. RUSSELL ON THE THEORY OF> DEFINITE INTEGRALS. 
We may extend this process, by performing operations with respect to the quan- 
tity (jU/). Thus we may operate on any of the integrals we have obtained by such a 
symbol as where F is any rational function ; and if it is an entire function, we 
have merely differentiations to perform. If it is a rational fraction, and the factors 
of the denominator are real and unequal, we may decompose it into simple rational 
fractions, each of which may, in its turn, be transformed into a simple integral. If 
we apply this operation to any of the results we have obtained, we immediately have 
a definite integralj J ..Pg'"'^F(Q) expressed in a series of single integrals, 
where the integrations are performed with respect to (jea), and {(jij) may be taken be- 
tween any limits. But (p) must in no case pass through zero, as the definite inte- 
grals, on which we operate with respect to {(ju), cannot be found for that value of (jij 
by the processes we have been investigating. There are many other operations of a 
similar nature, which it is easy to imagine. 
I am now come to the second part of this memoir, the investigation of those new 
methods of summation, and of the definite integrals corresponding to them, to which 
I have before alluded. Let us consider the series 
/3(/3 + l).1.2+/3(/3 + l)(,3 + 2). 1.2.3+^^-’ 
where (|3) is an integer. The following integral is known : 
Hence we find for the sum of the above series. 
Next let us consider the same series when ((3) is a fraction. We have 
r/3 /3-1 
d^dvv^{\ — vY *g®'“®cos(a sin ^)g’''®. 
r(/3-|-n) 7r«’ 
except for w=0, when 
and we find for the sum of the series. 
