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MR. W. H. L. RUSSELL4ON THE THEORY OF DEFINITE INTEGRALS. The first series we propose to consider is the following:- 1 1 2 r x2 3 5 ST 221 ttana6+ 72 9tan-, + tan 1+&c.16 tan 1 2Z 9 X22 25w2 L +g2 Put Tx2, then this series with its sign changed may be resolved into the three fobl- lowing: __ __ _ X2 1 9 2 1 2(1+x9) tan2+ 2(9+x2 )3 tan32+2(25 +2)5 tan + __ x 1 _ tan+- 5-tan _ + - - tan et 4 (1- ) 2 4(3-x)3 an3.2 4(5+x) +c 4(1 +s) 2 4(3Px)3 3.224 (5+x)5*t 52+& The general terms of these series are respectively, _ _ 2 1 _ -. tan 2{(2n+1)2+x2} 2n+1 tan 2(2n + 1) a; 1g_ _ 4{f(2n+l)-x} 2u+1 ta2(2n+l), {2+- . tan 4{(2n+1)-}
- 2n+t
2(2n + 1) These terms become after transformation,since Joodz(eaE~) 1 a -- =-tan-' Xo o2fl#s7l)U7sin dz ?-OVcs) 6-(2tl+1)ds2....1d -40aus XU (7rrZ -r r) v- - jav(e-os~ 4p~7 ss 00 00~f0 +)2~d. i Each of the series is consequently reduced to a geometrical progression; wherefore, sumlmingthe three progressionsand takinagthe aggregate, we have /o bo00 o 4 v2 -- -2(2 Vin (u 2n-)dsdvdudz JO-(2~io logilo 2e 2 /4- ~ >2Z2 2~~ 0 00 Jon 72 0 (00( - (W n 2 ) -00 d(0 (2, v2 2I' / 00duIM Kv (sCsVt_4
