Page:Philosophical Transactions - Volume 145.djvu/194

 MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 175 Next let us consider the series 1 1 cotrz2+ 22 cotn4 +2 cot +&c. x2 424 2X Xf /8IX* ?- 2. Let each term of this series be transformed by means of the integral -1 =X cot &,,X, and we have ioso21O1 1 dsd ' 2 ? gS_ - log'- - (2sin xu+ x" )dsdvduz 6OJO1_00t? (z - ) logs Z(1-eU X 2iogzs) ={(:2 y cotrx} 1 1 70x3eX 5 1rX &C Again, I secc--2 7P6 sej-+ 2 25sec -&c. 1-6 ~ ~ 1 t-$2~6 +S-7 5 = X{(~ :e ) -sec2 24' . Here we reduce each term by means of the integral ?0dzea- ~ez) 1 a r =z(? +')-- see * and we have noo ~ ~ v?o Aol OM rol ' XU) J(61 + Cos v) 6`47r2Z log -(2 sinxu+ dsdvdudz

S - (?7r0 + ?-7) ZZ1+2 TZ2) 4l 2 te2+ 2) 1 1 Also, since cosec vrx2 2 cosec 29 cosec & &c. Z*2 4x9- 9X2 r{( 2S=)2-cosec2X} we have, transforming each term of the series by means of the integral f0 za-ldz 1 + 2=cosec a2