1881.] certain Definite Integrals. 451 



dQ— r -——mr (1/8). 



Jo sm 2 v ' 



\* de (2n-l) sin0-sin {^-1)0 _ iy (1 } 



Jo 2sin 3 V J K J 



L 2 sill 2 110 71 — 1 ) 7T HftO^ 



f • • • - (181). 



f 71- 7/1 n 2 sin 2 



l de s 



psin_%0 

 J 0^m~0 



7/1 (2^— 1) sin6> — sin (2n— 1)0 , ... ziuo\ 

 Jo** *sin 2 fl J -=n(n-l)r (182). 



f 00 dQ n 2 sin 2 fl-sin 2 n0 _ n(n*— 1) F ^ ' 



Jo * 0sin 3 3 '.' • 



I* d0 ^~ com ® GOSn ® — n7r (184i) 

 J ~^0 2" 1 ; " 



From integrals (178), (181), (184), we obtain the three follow- 

 ing :— 



J-a, . (/(i)-/ W)+(/(i)-/(^')) =4 , / , (1) . . . (185 ). 



r" de (/.d)-/(^))+(/(i)-/(^ 9 )) =2r/j(1) .... (186) . 



Jo <?sm6> J K ' v y 



(/a)-/(^^^)+(/(i)-/(cos^)) =?r//(1) (187) 



j 2 ^{ 2 ^/(cosa?6^)-6- 2 ^/'(cos^6-^)} = 2^0(l) ...... (188). 







2 ^{6 2 ^( 6 ^sin6>)- 6 - 2 ^/( 6 -^sin6»)}=^{6?0 6 T + 6 -T0e?} . (189). 

 ^o 



-7T irr 27T iir iff 



\ 2 d0{e^f(sin0e^) +e~™f (sin e -*?) } z^ ^ + e ~ T 0fZj . . (190). 



J o 



In the three last integrals 0(«) has the same signification as before. 



Let p=a + b cos + ccos?0+ . . . +cos w 6> — 



vol. xxxii. 2 I 



