176 MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 
• CO /•OO /•OO ^ 1 1 
®“4^2iog£^ logs”""^ “ (2 siriOT + s-^^ — s~^^)dsdvdudz 
/%(Xt /•OO ^00 ^1 
J J J J 
%J 0 *^0 %J — <X3 %J a 
4<i:2 logga; 5 ) 
2 
—‘TT^l cosec V^— 
Let us next consider the series 
sin S I 2 sin 2fl . 3 sin 39 
l+a2‘^2H 
32 + «2 
^ gaC,r-0)_j-a(^-0) 
ea7r__„ S—CCTT 
The general term of this series is ; 
° n-‘ + cic-‘ 
also, we have 
and 
n^ + 
e sin Kzdz, 
sin sin 2 ^+£“^* sin 3^+ &cc.= - — 
1 — 2£~® cos fl + 
sin azdz TC ga(jr-0)_j-a(7r-6l) 
Hence i ^ r. ~ „ , 9 . n • 
Jo i — 2 £“-“cos9 + s“ 2^ 2 sin 
In like manner from the series 
cos 9 , cos 29 , cos 39 
aOCTT - — aTT 
^ _ ,r 
i + «2 ' 2H«^ ‘ 3^ + «® ' ^ £“’^ — 
2«‘ 
(£~*COs9 — e~^^)dz TT £“(’^ ®^ + £ J_ 
^ 00 
1 cos a 2 , ^ ^ o ^ 
Jo 1 — 2 £“®cos9 + £“^® 2 
cajT — arr 
Let us next consider the series 
1 
c7r_e — TT e27r.^s — 27r * e37r__e — 37r 
3^4- 
— &c. =— • 
47r 
ixr 1 C°° sin u.z.dz 1 £f^ + l 1 
We know that — -- 
J. 
sin 2m:zdz 1 
2ju. 
1 1 
— l 2 1 — g-2Ktt 4 4^^ 
• nn tx — m^ 
= 2 £- 
°° s\n ‘Imiz .dz , £" 
•TlTT — TOT 
£‘2trz_l 
2n7r 
Now a: sin sin 20-^x^ sin 3^— &c., 
^ sin ( 
l + 2x cos 9 + 
From whence we have xsin^— 24;^sin2^+34;*sin 3^ — &c. 
a; sin 9(1 
It is hence evident that 
'(1 + 2a; cos 9 + a?‘'^)^ 
dz sin 2ivz 
1 ,r/ 2,r \ r°° dz sin 2'WZ 1 s’" I . 4 . 
"—1)1 (j2:r4.2s^cos27r^+l)2(g2-^-l) + 2 (e^ + 1 ^^ 271 '?+ 1 ’ 
i 
«/£;.sin 27:2 
1 
“ Jo (e"" + 2£’"C0S27r2+l)2(£2»-~-l)~4£’"(£’"-hl)4s"-l)l 27: 
:2^_ ] 
