MR. W. H. L. RUSSEL]. ON THE THEORY OF DEFINITE INTEGRALS. 175 
Next, let us consider the series 
1 
1 A. 2 I ^ i- I ^ 4. I C 
— COtT^^+^^COt— +-^COt— +&C. 
4 x'^ 9 x'^ 
gTTX ^ g — TTX\ 2'| 
=g-| cotVj?— 
e — TTX 
Let each term of this series be transformed by means of the integral 
1 =-rcotaT, 
Jo 1 -^ 
and we have 
. 
v'o c/0 1/— 00 tJo 
£ “■*'® 4.;2iog£« logg"^ - (2 sin XU + £■*■“—£-«'“) dsdvdudz 
j)2 
[z — z^] logj 2’(1 — £'"“~44:nog,*5) 
■nx I ff— Tr.rx 2 
iKX __ - — TTX 
— cot Va; 
Again, 
1 i:x 
sec 
1 
3 wa;® 
sec 
sec 
Tra? 
.2 
•J?" 2 
1 ’2 9 g ' ^ 2^25 ^ 10 
■&C. 
? V_sec=- 1 - 
“161 V — 2 I 
^£2 +£ 2 ’ 
Here we reduce each term by means of the integral 
and we have 
<^£r(£“* + £-“*) 1 
• 00 ^00 ^00 
(e” + cos v) e~“~ 4 w 2 « logj"^ - (2 sin xu + £'»^«— £-■»•«) dsdvdudz 
^00 ^00 ^00 
J J ' J 
j/2 
"iy-T-ir _, 2 \ 
TT^ .ItX 
= i- sec ^-2 
(jTTZ ^ g— ^ g 27r;i!2;5 ^ 
2 
/ -VI 
— [ TTX ] f* 
VgT^g tJ j 
Also, since 
1 o 2 Tra?' 
— -cosec ‘zrxr—-^ cosec 
X 5 
x^ 4 
4 a;® 
3 T^X^ n 
. cosec &c, 
2 9 3 
9 x^ 
•cosec V4: h 
we have, transforming each term of the series by means of the integral 
C^z'^-^dz 
I -j— r — ^Tcosec KT, 
_"n ■*■ T ■3' 
2 B 2 
