1880.] 



certain Definite Integrals. 



103 



hence we obtain at once p + r, and q + s, which will render the solution 

 of the equations easy, at all events, in all possible cases. 



I would moreover remark that the validity of the process depends 

 on y/ (p + qx-\-rx 2 + sx s ) expanding in a converging series, so that the 

 method of evaluation here, a. of course in (109), (110), depends on 

 certain conditions, to which the constants in the integrals must be 

 subject. 



II. " On certain Definite Integrals." No. 7. By W. H. L. 

 Russell, F.R.S. Received January 6, 1880. 



By a development of the methods indicated in the former papers 

 we obtain the following integrals : — 



r— 



(y\K\ U dQ cos ra cos (n — 2) + a co s" +1 cos (n— 3)0 

 J sin 2 0+(l + a )2 C os 2 



_ 7r f n a. 1 



~ 2^ 1^+2 (a + 2) 2 J * 



(116.) 2 d0cos 2 0e2* cos2 C os (ajsm20)=-(aj + 2)e* 

 J 8 



^ J o 1 + 2* 3 cos 3 cos '69 + « 6 cos 6 



_ 7T f ft 3« 3 1 



~2^l^+8 (a 3 + 8)2 J * 

 (118.) j J d0 COS 2 e acos3 cos30 cog ( a cog 3 s i n 30) = |^2 + ^€*. 



1 J o _ sin 2 0+(* + l) 2 cos 2 ~2^^+2' 



n20 v f| 7 cos^0 cos/a0 + (a +|6)cosP + *cosO»— 1)0 + a/3 cos^ +1 0cos(/*— 2)0 

 ( ' ; J (sin 2 + (a + 1)2 cos 2 0) (sin 2 0+ Q8 + l) 2 cos 2 



2/-i(* + 2)(/3 + 2) 



This integral may be written 



7T 



(121 j 2"^ cos,x 61 cos ^ cos^ 4 " 1 cos (fi— l)0 + gcos^ +2 0cos (/*— 2)0 

 ^ J sin 4 + (^ 2 + 2p — 2g + 2) sin 2 cos 2 + (p + 2 + 1) 2 cos 4 



2m-1( 2 + 2p + 4) 



