174 
MR. W. H. L. RUSSELL ON THE THEORY OP DEFINITE INTEGRALS. 
The first series we propose to consider is the following 
2 1 
X 5 
9 
5 
6 
25 x^ 
I , ■irx'^ , 3 ^ t:x^ . 5 ^ is 00“ , „ tt | / £ " — £ 
tnn 2 “h ^2 ^ ta.n tnn i -l-&c. — 
TT.r 77^ \ 2 
“T \ 
16 1 \ — TTX J t3.ll ^ 
]■ 
Pat ■7rx^=§, then this series with its sign changed may be resolved into the three fol 
lowing : — 
■ §. 1 g 1 g ^ 
2(l+x^) '^^*^2“'" 2(9 + «^)'3^^”3.2^" 2(25 +«2) 5 5.2^"^^' 
X g 
tan: 
X \ , q X \ q 
tan_^+ jT^— . tan ^+&c. 
4(l-/r) 2“'“4(3-^)3 3.2"'"4(5-;«’)’5 “ 5.2 
X 
- ^ g X \ g a?l g . r, 
4 (TT^ tan 2 — • 3 tan ^ ^ g • tan ^ d- &c . 
Tlie general terms of these series are respectively, 
x^ 1 g 
tan r — ^ 
2{(2n + lf + x^}’2n + l’^2(2n + l) 
X 1 g 
tan 1 ^ 
4i[[2n+\)—x] 2n+\ 2{2n-\-\) 
X 
1 g 
tan 
~4:{{2n+\)-x)'2n + \ 2(2m + 1)' 
These terms become after transformation, since 
j: 
g — 2 
ij duz ‘^”+^^“sin4?Mj dv {z'’ — ^ ^ 
( 2 «+l)«^ ds , 1 2 „ 
J-s 
’5 S 
Each of the series is consequently reduced to a geometrical progression ; wherefore 
summing the three progressions and taking the aggregate, we have 
rs 
0 c/Q —oo%d 0 
t)2 _ .1 J _ 
(£"— cosw)£“““w«logj -(2 sina;M + £'*^“ — £ ™)dsdvdudz 
- £-2“-2-5is2) 
