166 
MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
From hence we obtain, using- the first of the two integrals and the series 
*+!(‘+l:l'r^+it-6-T^3+&o-=s(^-2).'+4^+2) 
lil 
and also 
cos^^ cos (jM/ 2 sin ^cos ^+4^— tan — 2)£'"+^(w-+2), 
fj/^e 
{"*00 /'TT fJLZ cos 0 
rr 
Jo J-Tl 
z cos(c sin ^)cos 
fjiz sin I 
■30 
WC^ TTC^ 
The second integral will require in its applications, that we equate possible and 
impossible parts, in other respects the results will be analogous to those we have just 
obtained. 
There are one or two other methods of summation which I shall briefly notice. 
We see at once that 
. &c 
^^2^1.2 3^1. 2. 3 — 
TT 
1 2(— 1)’-“' r'2 
Now if (r) be any integer, -= 
P- 
COS 
Hence 
Whence 
The integral 
can be employed in the same way. 
Again, 
TT 
^ rf^logjCos 0 pQg gjj^ 2 0—2 0 )=^-^—~. 
si 
Jo 
sin 0 cos^’’ 0= 
2r+ 1 
whence 
2 . . j. ^ 
COS” 0 cos n 0 
TT 
£ 
d0 cos”' 0 
^ 1 Wr^ 1 -{- S ^ 
Hence using the series 1 + ^ — 34 +^^.= — ^ — j 
2'^ 2'2‘^‘^ 
we find 
TT TT 
cos ^ (pe 
§ ^,|u,2cos20cos!pcos(20 — (5) 
cosi sin(2^— cos^^ cos 9+tan 
