MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 173 
same factorials, so that we can deduce the value of many definite integrals from one 
series. 
I shall now give an example of the summation of a factorial series of a somewhat 
different nature. 
Consider the series — 
S/ C(^ 
^ + 2^ (a® + 2^) {a^ + 4^) (a^ + 2^) (a^ + 4'’^) . . . («^ + 2®w^) 
OTT __ air 
we know that P g"^(cos^)"= 2 , o2^/ 2, 02 
Hence by substitution the above series becomes 
f2 
-J—H dtfl I + &c. 
_ajr\ 1 ^ 1.2 1.2. 3. 4 ' 
^ ^ ^ f ^ cos 0 _L ^ cos 0 
a7r\ 1 
eT_s-T jj - 
^^ga 0 { 5 V:.cos 9 _j_ 5 - 
There are other series of an analogous nature which may be summed in a similar 
manner: the object of introducing the above summation in this paper, is to point 
IT 
out the use of the integral g“®(cos when impossible factors occur in the deno- 
minators of the successive terms of a factorial series. 
In the ‘Exercices de Mathematiques,’ Cauchy has proved that if be a quantity 
of the form ^(cos sin <p), and z<p{z) continually approach zero as ^ indefinitely 
increases whatever be (p, then the residue of (p{z) is equal to zero, the limits of § being 
0 and (co), and those of (p, w and — -r. From this theorem he deduces the sums of 
certain series, which 1 shall presently consider ; but must first give certain results 
which will be useful in the sequel. 
V -1 
Since 
2 cos 2xdx=- 
-1 V~a T” -21^ 
^ 0 ■ c d. i 
2 V ' TT J-co 
Again, since 
r 
J-00 \/ a 
we find 
1 . /~ C* 00 
— V a k « — — 1 
2“ = = 1 2 * a 
2 Vtt J-x 
whence we have 
p dx{f 
2 -V^TT J-00 
)g"T- 
2 B 
MDCCCLV. 
