1881.] certain Definite Integrals. 333 



j r ^cos2 0{/ 6 cos5 ^ + y 6 oos3 5 e- 3i ^ = ^|2/ 6 i+^.y t ij . . . (138). 



j* dO cos* ^«^+A««-"*}«||/e»+^/v} (139). 



f" d & irx f(2 sin %e<r+*)) + e ~^f(2 sina^Cr**) ) 

 Jo 33 (a 2 + »*)(&* + a5*)(c* + a> a ) . . . (e 2 + z 2 ) 



a(a 2 -6 2 )(a 2 -c 2 ) . . . (a*-e*) J k ' ' 



4- - /( 6 -2J_l) . e -26r + (140). 



Similarly we may find 



P° r7 e 2 ^/(2 sin a e* (r+*) ) — 6- 2t >*/(2 sin a e~* ) ) 

 Jo (tt 2 + a 2 )(& 2 + z 2 )(c 2 + a 2 ) . . . (e 2 + ^ 2 ) 



[j; r /(2sin^(^) + ^/( 2s in r -(^)) _ 2;r/( _ 1) gf 



Jo l-2«cos0 + * 2 l-* 2 ^ ; ' 



Similarly we may obtain : — 



Cn e^fl 2 sm«6*(-2- ) )-6- i ^/( 2sino6- / ("2- ) ) 

 dO sine V - I V - Z .... (143). 



JO 1 — 2*COS6> + a 2 J 



[ a> ^ irtt *f( 2 sin aa*' (t +ax )) + e-*>"*/(2 sin aa'6 -'(i a 0) 

 Jo ' (« 2 + * 3 )(i33 + a5 3)( 7 s + aJ 2) ...(tf +fl P). 



cos aa: 



e?^/(2 sina^'(T +a +e-2«'««/(2 sina^Cf^)) 



+ aj 2 ) . . . (\ 2 -f as 2 ) . sinaa; 



(144). 



(145). 



Jo' ' " (* 2 + ® 2 )(/i 2 +a3 2 )( 7 2 

 fl^ f(cosej°)+f(cOBee-») _ir f X ....... (146)> 



J «3 2 cos 2 + a 2 s i n 2 ^ ax* x + a 



-f^ ^mif (1) -/(0)) (147). 



jo 33 . 



These formulae may be greatly extended. I add a few examples of 

 their application to particular cases : — 



^.-'■■^(crf^jg^ 



J <B»cos»0+<i»Bin s 2«s 1 ; 



