Page:Philosophical Transactions - Volume 145.djvu/195

 176 MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS& C CE +14X21og&Z log,~ V(2sinxuzus =dddd 200 AM 0041 v 2 1 (z+z ) log,2Z(1 +?4,21og6zS) .; Tr {cosec2Trx- ( 2 2-)} Let us next consider the series sin0 2sin20 3sin30 Sa(V-0)_e -0(V_8> _+ __ _& C.=-. 1+a 222?+ a2 32+ 2 2 so7-C.- The general term of this series is in +o also, we have n+:= f1 sin zdz, and ZsinQ+2 sin 20+s-3Z sin 3d+ &c.=1 esin 0 1-2e-r cos 0+ e.2z- r 9-Zsin azdz ir t-) -E-(7r-0) Hence JO12- --- coso+e2D 2 sin0o s-e Mr In like manner from the series cos0 cos20 cos 30 IC-0)+8- 2+ 22+a 22+ & 2.= ar 1+~2+'2 32+ ~ 2ca COSc-z1 - 0 - 6-2 --)dzi saO-0) -a(- ) I Cos zcs~rzdz ___ ___ _ JO_ 2e- ZC0os9 +S-2Z 2 ofr_ ae 2c Let us next consider the series 1 2 3 1 e7r-eg7r Z2v 2r+ 37r - c...& c.4 fCsinUz.dz 1ep 1 1 We know that -V j? sin 2na-zz 1I I I SO sirz_ 1 2 1 -e2t"r 4 4n7r 1 (0 sin 2nwz.dz e-r c-nlr enff_ 27r,-1 ?2 +2n Now x sin 0-x' sin 20+x' sin 30-&c., xsin0 I+2Xcos0+X From whence we have xsinO-2X2 sin 2d+3x3 sin 3d-&c. x sin 0(1- 2) (1+2 cos0+ 2)2' It is hence evident that 7r2( 1) rm dzzsin2rz2,rz JO(2+ 2cos22z+ 1r2(e2-21) +2 (e?1)- 1' dz.sin2rz 1 fe291 1 J (27r +27rcos27rz+1)(12 (1)427r(eT+1)2(.772 ) 2vr