452 Mr. W. H. L. Russell on [June 16, 



\*de{f(j**»)+f(p^°)}=*fL (191), 



if p=a + b sin 0+ . . . sm n 0, where n= 4m + 1. 



\ F de{f(pe^)-/(pe^)}=^fL. . ; (192). 



Jq ^ 



Let x, /3, be the roots of x 2 + 2x + 2=0, then we have the following 

 integrals : — 



^dx{e^f(e^)+e^f(e^)}=(P(l) .......... (193), 



^dx{e^f(6<^)-€^f(e^)}=i(/)(l) (194). 



We shall also be able to find : — 



r^rx^ (/*)-/(**)) ( 195 )- 



Jo ^ + 2to + ^ 



r^zlr ! X- 2 ^( 6 "> + /( e "")) (196). 



Jo # + 2M; + a; 2 



Let (p r (x) = fff . . . f(x)dx% then— 

 [^^{6^(cos^)-6-^(cos^)}=2^V0 2 (^ . . . . (197). 



[ 2 ~^{e 6 ^/(cos0 G ^^ .... (198). 

 ^o 



f ,r 6>^/rsin6>) = ^f 7r ^/(sin6>) (199). 



Jo v 2Jo 



\7t { {^- /o } • (2oo) - 



dOcosr0< -— / -— + — /— } 



JO le-* 1 -* e-« ! -« e ft -/ ^-«J 



where A A X . . . are the coefficients of the expansion of f(x). 



