MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
163 
The following- are instances of the application of this method obtained by using 
series 1., III., IV. : — 
(a[jb sin cos {(JbZ sin 0) 
-TT 
COS 9 + ;3 cos (p +/u,«« COS (6 + cp) (1 — 2;)“(1 — 
1 f*l i*7r 
0 «y0 — TT 
cos (a sin ^)cos(|(3 ^\t\<p)co^{^(JjV% ^\x\{6-\-(p)'^dM<pdvdz 
277r2 2^2 
^ (3->^ |tA05^ — 2)£ 
27r^ 
^ 2 
20 81{|x«/3)3 '' ^ 81(ju.«/3)^ 
n TT ^ 
do dv v{\—v) 2 £^“+'‘"5‘=‘>"®cos(2^+p?; sin ^)cos (a sin 
-tt 
— 2 V(3C|U. 
Again, we know that 
7 
r 
dOcos^O cos 
ffr(/3+ 1) 
■2»«r(^+i)r(2^+i) 
from which we may deduce the following: 
TT 
£ 
cos“+* gC« ^'>^^d0= 
‘jrT{a + b— 1) 
2“+*-T«rz» 
Now consider the series 
1 -l-%-L. ^(« + l) I «(«+l)(« + 2) , o 
^/3 ^/3(/3 + t) 1.2^/3(/3 + l)(/3 + 2) 1.2.3^ ’’ 
where (a) is greater than (3. Then by the above formula 
]^=-^r(«-/3+i)r Scos’^'-i 
2 
and we find for the sum of the series. 
2a-i r/3r(a-/3 + l) 
TT Ta 
TT 
£ 
COS*"’^ * l)j« j2cosfl 
In like manner we can find the sum of the series 
I _i-“.“'r_L«(“ + l) «'(*'+!) _L Srp 
^/3 ^/3(^+1)’s'(^' + 1)T.2^ ’ 
where a is greater than (3, «' than /3'. 
