60 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



other than the solution v = function of u alone. When the variables are taken to 

 be x, y, u, the equation to be satisfied by v is 



d*v . d*v 2 I dto , d*v \ _ o 



cly? dy z A \ dx du dy du t 



where 



A = 1 + xp + yq = 1 + 17. 



Now 



so that, as 

 we have 



dv r f . , A r 



^ = a, + to + y? > ^ ' 



also 



^ ,ar 

 ^ : ~-P a, ' 



and therefore 



Hence the equation becomes 

 and therefore 



2 er 



~r ^ ^ 0. 



1 + r, dr, 



where ^ and / are arbitrary functions. We thus have the theorem* 



* This result can also be expressed in the form symmetrical as regards the three variables. When 

 thus modified, we have the theorem 



If Pt ?> '''> 6e three arbitrary functions of u such that 



p a ~ + <f + r-2 = 0, 

 and if u be determined as a function of x, y, z, by the equation 



au = xp + yq + zr, 

 where a is any constant, then, writing 



a xp yq zr 

 where F and G are arbitrary functions, v satisfies LAPLACE'S equation 



V2 = 0. 

 This theorem also was stated in the paper referred to ( 33, note). 



