1877.] certain Definite Integrals. 509 



(31 .) J dde 2d cos (sin'-' d sin 20) = 



(32.) r c?0sin0esi°'^«os20sin(siji2Qg|j^20 + 0) = -47-. 

 J 2v e 



(33.) r ^ c?a0{e« 20 sijj 2(0 - a cos' 0) + e-« 20 gin 2(0 + « cos' 0)} = 



^ 



r *^ (a? — c{'^^)cZ^ _ e^"--\-a 



Jo ((H-« + fOsma^— asm3aii?)(l + .:^^) ~'^e2'^-a * e2<^-l' 



^ '^Jo ((l + a + a')cosa^+acos3a^)(l + w')~"'^e2* + a*e2'^+l' 



(36.) 1 C?0e^cos0cos(^sin0 + 0)log,(l + cos0) = '^^^ 

 •J 



TTSma 



J I cg0 log (1 + 2a cos "0 cos nd + a' cos 2^0) ^ (a4-5)«4-aa" 

 ^ a' sin' + 6' cos'0 ~ ab (« + &)» 



I observe that in (28) ^ and m must be odd and even or vice versd, 

 and that in (29) n must be even and m odd. 



The integrals here are not solitary examples : each of them may be 

 considered as the representative o£ a very numerous class. I have selected 

 as examples those which seemed most suitable to be brought before the 

 Eoyal Society. 



P.S.— The following were added 13th February, 1877 :— 



f , . . / . 30\ TT/^-l 1 \ 



(38.) J ^ 0c?0 cos « sm (^^ sm 0+ cos - = -{--^ + — j . 



(39.) p,,, Bin20 + (l-.)sin0 ^ r (,..^l)log.(lHh.)-.| 

 ^ '^J^ 1 — 2a?cos 0+a;' [ cc^ J 



