1880.J certain Definite Integrals. 



(104.) cos cos (9^0= cos sin Odd 

 Jo Jo 



=7r /i_JL+_i_ L- 1 L+ 



W l 2 2 ;]* . 2 2 " 2 4 l 2 . 2 2 . 3 ' 2 6 



(105.) r 6 cos ^6>= r e sin ^6> 



Jo Jo 



- 7r /i + 1 + 1 JL+ L_ _!+ 



' 1 2 2 l 2 . 2 2 ' 2 4 l 2 . 2 2 . 3 2 ' 2 6 



In the same manner we may find 



e COii0 dO 



(106.) £ 



lo (l-f-2*cos<9 + « 3 )i 



(107.) cos r &£0 V a + 6 cos 6> + c cos 2 6> + c cos 3 6> 



= 7 r(A A / . + A 1 A r+1 + A 2 A J . +2 4- . • • ) 

 when Aq, A 1? A 2 . . . are the coefficients of expansion of 



\/p + 5* + r^ 2 + s« 3 , 

 r, s being determined by the equations 



_p 2 + 2 2 + r 2 + s 2 =a + ^, 2_pg + 2gr + 2rs = & 



2pr+2 2 «=£, 2pg=£ 

 We are able of course to deduce 



d# cos r + 5 cos + c cos 2 + e cos 3 0) , 



<io8.) r 







and similarly the more general integrals 



(109.) ^ do co& r 0<y(a+bcos0+c cos 2 + . . . ccos"'6>), 



and also 



(110.) j 77 d0 cos r M %/ (a +b cos + c cos 2 0+ . . . + ccos M 

 Since the following integral is known : — 



