20 



Mr. W. H. L. Russell. 



"On Certain Definite Integrals. No. 13." By W. H. L. 

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



In a paper which will be found in the " Proceedings of the Royal 

 Society " for June, 1865, I gave methods for expressing the sum of 

 certain series by definite integrals, or in other words, of expressing 

 F(a?) by the form fPQ x d9. As shown in my last paper, this method 

 is immediately connected with the solution of those partial differential 

 equations which have constant coefficients by definite integrals, a 

 circumstance which never crossed my mind til] lately. In the present 

 communication I hope to make further extensions in both these direc- 

 tions. 



Case I. It was proved in the paper cited that the function 



could be expressed in the form f~PQ n d0, whereas q>(n) and x( n ) are 

 rational (misprinted identical) functions of (n). In the same way we 



may obtain <fi(n) + V ( x n+ Z/a(n)). For it was proved in that 



paper that V ((fin-\- Z/x n ) can be expressed in the above form if 



1 s_ 



e7 ' ( " , (x( n )) 9 can ^ e thus expressed, and therefore 



</ (pn+ V( x ( n )+^/u{n)) 



can be thus expressed in the form fVtydO if 



i 



can be expressed in this form, which can be done by repeating the 

 process. 



This investigation assumes, however, that x(. n ) + ^ w(n) is less than 

 unity. 



Case II. Suppose it were required to reduce e N , where N" = 

 v/00) + V X {n)+V~^(n) to form fFQ n dO. 



Then 6 N = ir^c s( S m^-g) and gince ^ denomiliator can 

 7tJ 1-2Ncos0 + N 2 ' 

 be rationalised, we fall back on Case I. IN" must of course be less 

 than unity. 



Case III. When jp is greater than 1 



F 1_ P 3-1| V Wt + Vr" 

 p 7T J 1— 2pcos0-r-p 3 



and p 2 -l=p 2 -2pcos6> + l-f-2(p- costf) cos 6>-2sin 2 6>. 



