MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
10.3 
Now put ^= 2 ? then 
— 1^.-? p 
r<i ' Vi, X 
COS® ‘^6 cot® d S®‘® ; 
and putting’ we have 
J. 
whence we find, from series IV., 
T{n+2) __1 ^ ^ ^2 C0S”+® ^ COt®^ 
2 . 2.3 
' 2.3.4 
2 2 2 
3 
.i^-+&c. 
. p* 2 /^.r 50£ 
■ 1 I COS®^ COt®^ g 2 gCos(!) jjQg gjj^ ^ Z^‘'^d(pd 6 g»4Cose gi{fl + 5!)^ 
•^0 fj — TT 
/ • Stt “T ^2 , V2\ 1 
.’. (since— s*= o“+ — 5 — ) we have 
\ 3 3 3 / 
IT 
j j cos®^ cot^^ 2®“'^ COS (sin cosjjM/ cos ^ sin (^4'?') + '2 
Let us again consider the series 
1 + «(«+!)(« + 2) ^ 
^■t"/3^^^(/3 + l) I.2"h/3(/3 + l)(/3 + 2) 1.2.3^^^^* 
Then making use of the integrals 
r(a+w) = r c?x, and r(a4-w) = A“'^”g“¥^“'^”^ T 
Jo Jo 
where (A) is a constant quantity, we find as the sum of this series, 
and 
r/3 
Fa' 
r« 27r 
g ^00 ^ 0 
^Jo J- 
dvdz 2 “ 's * 
(l +w)^ 
2 2 
f.f. 
;s“ *2*'“ 
1 +w 
(1 + ivy 
g i+iB ; 
also when |3 is an integer, we may find the following expressions as the sum of the 
same series : — 
FiS 1 F” 
Fa'wci^i j X““'2“®COS (csin^) 2®®°®®.2^^"‘^‘®.2 ^ > 
and also 
0 ^ — TT 
iricx. 
’i L 
dddz ;s““‘g*'* COS (c sin ^)£®®°"®.2'^“*'*®.2 
2 A 
ihxxt 
MDCCCLV. 
