Page:Philosophical Transactions - Volume 145.djvu/184

This page needs to be proofread.


MR. W. H. L. RUSSELL ON THE THEORY OF DEFINITE INTEGRALS. 165 Now put b=2, then 2a - 7rg 2 a cos-2 Cot' lsOia- and putting oa=n+?, we have ( G?)= v- 4X cos'Q+0cotwOgOi(l+s)n whsencewe find, from series IV., 1?~- ~ ?73+ 2 1i G2 + &c. 2 22 3j 'i (7r5i =2 I 4 cos COS cot-20g2~sgcos CsSin @ g2zxdpdO gcosO ?i(6+0) 2~~rJ2 . . c(since 2 f 4 serieswe have X j'wcos-ff cotO ? cosCO (sinp) COs6COB(0+dd cos{it cos Osin (O8+) + 2 + 2 } O 73P - 7 -3 .4 -2 72+ i2 (1-g2 -2 Vvp+)- . Let us again consider the series c7 r+1) x.2 we(ez+1)(++2) hav3 Then making use of the in-tegrals r( +n) dz, and 2 +n)=+ 2 where (h) is a constant quantity, we find as the sum of this series, an~d r~*2h ^a 2C dvdz za1kiz (l+ T" ~v; also when i0 is an integer, we may find the following expressions as the sum of the same series a rl 10x00 'r XZC M-l ddzZOC1~cos(csin0) ecosO ((31)iO e r a2r (I +r vih0s and also ra , - h dhdz ki c (I+csin Io (3l)o MDcccLv. 2A