104 On certain Definite Integrals. [Jan. 22, 



r— —l 

 (122.) I 3 dO e* cos2 * s i n ( a S i n cos + ,u0) =—. 



J sin0 2 



(123 } f^cZ0cos0^ COS ^^*~'~ C0S ^ a ' n ^ sin sin tan ir 



^ ,J C ° S (l + 2/3costan6' + /3 3 )(\ 2 sm26? + / i2 cos 2^ _ 2^.^( 6 + i3 ' 



(124 



P" 1 0d0i 



Jo(i+^ 



0^0(1 - a 3 cos 2 0) sin sin -i 



(125.) £ 

 (126.) J* 



a 3 cos 2 0)^l+£ 4 cos 4 »\/2 1+^ 3 



^ . x %l 



1QQ . . a; 3w cos 3w 



d06 sin 



(a 2 + a 2 cos 2 0) . . (e 2 + x° cos 2 0) ^ J 



cos r0 



-dO 



(1 — 2a cos + « 2 ) (1 — 2/3 cos + /3 3 ) 



?r_ / _ \ 



(1- ll-* 2 1-/3 2 J ' 



(127.) Hence we see the valnes of 



J; 



d9 . (f<fii+fe-*i) 



o (1 — 2« cos + ^ 2 ) (1 - 2/3 cos + /3 2 ) 

 By a similar method we may find 



dO cos rO 



o (1 — 2acos6> + a 2 )(l — 2;3cos0 + £ 3 ) . . . (1 — 2/*COS0 + /* 2 )' 



(128.) J 



Jo (X 2 cos 2 + ^ sin 2 0) sin X 2 X V V + V ' 



(130 [* de ■ 6aC ° s3esin («sin2fl + 0) _ *- ea ^Qt-Xj ^.M-A 

 ^ ' ; Jo (\2 C os 2 + ^ 3 sm 3 0) .sin0 \ 2 X^ju, 



(131.) We may also find 



d0 sin (2r + l)0 



[" 



J o sin 6>( V sin 2 + ^ cos 3 0)(\ 2 2 sin 3 + ^ cos 2 0) . . (X w 2 sin 2 + \ w cos 2 /3) 

 non ^ [ n sin r0. dO ira. r 1 / r 1\, 1 + a 



(132.) "^-Tt -5T-( »'-— ) lo g«:j 



when (r) is even, with a similar expression when (r) is odd. I shall 

 now hope to prove that every function of an algebraical magnitude 

 may be regarded as a centre, from which systems of definite integrals 

 emanate in all directions, like rays from a star, in such a manner, that 

 the value of each integral is equivalent to the original function trans- 

 formed by a known symbol. 



