On certain Definite Integrals. 21 



Hence 



P— 1 - = 1 + 2 (PZL^l . C o S 



1 — 2pcos0 + jp 2 (p — costf) 2 + sin 2 6> 



2sin 2 



(p — cos0) 2 +sin 2 



f°° -z(p-cos6) f 00 -z(p-cos0) 



= 1 + 2 cos 6 cos z sin — 2 sin # I 6 sinz sin0c?6>. 



Jo Jo 



By this means can be expressed as double integral. So can 



F(j9), but then p must be less than unity. 



We will now apply these considerations to the solution of linear 

 partial differential equations. 



Let , — \u=0, or as we shall write it, F ( x— , y— )u=0, 

 \dg d-r]) \ dx dyj 



then taking as before a specimen term Ax m y n , m and n must be con- 

 nected by the relations F(m, n)=0. Suppose from this we find 



Then, as will be seen by the reasoning employed in my former paper, 

 the equation can be solved if 



e ,OSe v / 0w + y x w + ^)+ .... 



can be expressed in the form fPQ n dO, which brings us to Case II. 



The same process may in certain cases be applied to partial differen- 

 tial equations with three independent variables. Consider the series 

 A + B^+B'i/ + C^ + C'^ + C'y + , . . . when A, B, B' . . . 

 are arbitrary constants. This may be written on Poisson's principles 



F 1 (0)+F 2aJ . 2 / + F 3 (aO.2/ 2 + • • • 



when F l5 F 2 , F 3 , . . . are arbitrary functions, and this again 

 F(#, y) when F is an arbitrary function of the two variables. 



• • • • • cl/tt c(/^'Ll' 



Now consider the partial differential equation — =2 , or as I 



d £ dfjdrj 



shall write it ^«A^ft==2^»^-^y— and let Ax m y n z r be a specimen 



term of the solution, as in previous cases, then r=2mw, and our 

 object must be to reduce x m y n z mn to the form /PQ^Qg* ; this may be 

 easily done by remembering that 2mn=(m-\-n) 2 — trfi — n 2 , for 



| 6 du= y/ 7T 



