334 Mr. W. H. L. Russell on [Jan. 27, 



& dO log £ (1 + 2a cos 2 cos 20 + a 2 cos 4 6) _ tt ^ (a + ^3 + ^2 ^.q. 

 J .? 2 cos 2 + a 2 sin 2 ° ge ;0 + a) 2 ^ 



pto sin0 1 



Jo *l + 2acos0 + a 2 2'1 + * • • • - I 



r^^ cose ^(^) = ^-i) (151); 



Jo Z 



| Q y V y(l + 2acos0 +a 2 )-(l + acos0)=^(v / l + *_l) . (152). 

 In ray last paper I gave the integral 



^ shire ...... aa2) . 



j 1— 2acos0 + a 2 



It is obvions that by a similar process we can find 



(* d0 cosrO r- s jnr0_ 



J l-2asm0 + a 2 Jo l-2asin0 + a 2 V h 



remembering that 1 — 2a cos ^+ 0^+ « 2 =1 -f 2a sin + a 2 . 



Also since 



tocos(r+l)0_2f dOcosrO _2aX d0 cos rO _ tocos (r— 1)0 

 J (a + 6cos0)* ~ &J(<x + &cos0)' 1 - 1 Tj (a + 6 cos 0)" J (a + 6cos0) M 



. . . (155), 



to sin (r + 1) _ 2 f d 6 sinr0 2 a f 670.sinrfl _ to sin (r — 1) 

 J (a+ b cos 0) n ~~ b) (a+ 6 cos 0)* -1 ~£ J (a + 6cos0)» J (a + 6cos0) /i 



. . . (156), 



i tosin(r + l)fl _2f <?0cosr0 _2a f jg . cos rg + to sin (r— 1)0 

 J (a+6sin0)" ~~ b J (a + b sin 0) Tj (a + 6sin0) M J (a-h&sin0)'* 



. . . (157), 



tocos (r + l)0 _2af <70sinr0 _2f t70sinr0 to cos (r— 1)0 

 J (a + 6sin0)" 6 J(a + 6sin0) tt 6 J (a + b sin 0)'*" 1 J (a + 6sin0)« 



. . . (158), 



it is manifest that : — 



['ae (159), ['*>„ a siar0 n . (160), 



Jo (l-2«cos <?+«?)» V ' Jo (l-2*cose + * 2 )*< " 



