58 PROFESSOR A. R. FORSYTE ON THE INTEGRATION OF 



It is immediately obvious that one solution of this equation (and therefore of the 

 original equation] is given by equating v to any arbitrary function of u, a result that 

 admits of simple verification.* 



34. But it is not at first sight clear how this solution connects itself with the 

 general solution indicated in 31 ; the connection can be made as follows. 



The general value of v is 



dw dio 



if this is to be an arbitrary function of u, say /(), as for the solution under considera- 

 tion, we must have 



ihu chv . ft \ 



and consequently 



w ^ f(u) -f G (u, p'x + q'y), 



where, so far as concerns this relation, G is any arbitrary function of both its argu- 

 ments. Writing 



this is 



&' / / \ | /"N / \ 



w = j (U) + U- (u, r)), 

 so that 



dy? clij- 



/)~i fTf 



' "" " / / ^ O > 



' Gin" 



Now , 



p* + q * + i = 0, 



* This result was published in the 'Messenger of Mathematics,' vol. xxvii. (1898), pp. 99-118, in a 

 short paper entitled " New Solutions of some of the Partial Differential Equations of Mathematical 

 Physics." The form was altered from that in the text, so that it might be symmetric in the variables, 

 and the theorem was given as follows : 



If p, q, r, be three arbitrary functions of u such that 



p* + f + r a ~ - 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, also if v denote any arbitrary function of u, then v satisfies LAPLACE'S equation 



V 3 D = 0. 



