1888.] 



On certain Definite Integrals, 



311 



V. "On certain Definite Integrals. No. 16." By W. H. L. 

 Russell, F.R.S. Received May 31, 1888. 



In these papers I have considered incidentally the advantages 

 gained by differentiating and integrating with regard to the quanti- 

 ties which are independent of the leading variable. In the present 

 communication I enter upon this subject more systematically, as it 

 evidently admits of wide extension. 



J m* + x z m J am 



C n da 

 J J sm0 



(See No. 88 of this series.) 



{ 6 itan0 + gitang 

 : ~ : . r— - + 



-i (tan fl + 



e -itan0 -j 

 1 _ » 6 -itan-0 J = 



1 1 



1 — pa* tan ^ 1 — ^6-^ an ^ J 2 e — e — n 



This is obtained from integral 21 of this series, where, however, in 

 the denominators of the integral cos tan 9 is misprinted cos 0. 



f* r cosxe iitanx + x) / cosxe ix \ 

 J ' \ 1 — a. COS xe ix \1 — ot COS xe ix J 



+ cosa' 6- ;(tana; + a:) / cos xe-™ \ \ _ 77- _I 1 



1 + a COS #e _i * \1 — a COS iCe _ 'V / e c 2 — a 2 — a 



See integral 22. 



/•n 



-t— { 6*^0 (cos ^) + e- id (cos -*"«)} = jn0(l). 



o sm 



See Abel, ' CEuvres Completes,' vol. 2, page 88. 

 P- 



S cfa {cos 2 e 2 * 0032 6 toBja2 fl (2 cos 6> e 2 *) 

 • Jo 



+ cos 2 e e- x cosa 6 e - ix sin 20 (2 cos e 6 - *) } 



= tt(20(1) - 0'(l)e* + 0(1) . ,«■). 



See integral 116 of this series. 



