22 On certain Definite Integrals. 



foo 2 _ 



Hence e 2au - u *du=e a s/ir 



and therefore e im+n) °= * f 



V 7tJ -« 



_J_f 



e />* cos 2mpdp, and so for e " 2 . 



These transformations give the required form. 

 If we have two partial differential equations — 



/ d dx d\ A 

 \ dx dy dz J 



then substitute as before Ax ,n y n z r for u ; then we have the equations 



I\0, n, r)=0, F 3 (w, w, r)=0, 

 whence m=0(r), ?i=x(r), and we fall back on the first case. 



''On Certain Definite Integrals." No. 14. By W. H. L. 

 Russell, A.B., F.R.S. Received June 18, 1885. 



It follows from the expansion of cos w 6> in terms of the cosines of the 

 multiples of 6, that 



n — l n — 2 n—r + l_2 n i 



cos n0 cos (n — 2r)0d0, 



Jo 



and consequently this theorem can be used in the summation of series 

 involving binomial coefficients. I propose to give a few examples of 

 this. 



From the binomial theorem, when the index is even, we have 



r*" ^ cos 2 '* 6 sin (n—l)e cos ne_ tt f 22»-l_i_l_(2w— 1) . . (n + l)\ 

 Jo sm~0 W n \ " 1.2. . .(»— 1)\ 



and when the index is odd, 



f ,r JQ cos 2n+l e sin nO cos nO f , 1 "I 



Jo^ sn> =^{4-^1 



Since (1 +x) n ~ l = (1 + x) n (l — x + x 2 — # 3 + ....), therefore 

 equating the coefficients of x r } we have 



