70 PROFESSOR A. R. FORSYTH ON THE INTEGRATION OF 



a solution of which is given by 



x 8*F 



-- 



v = F (*, * ) - 4 -- 7 - -y, 2! Ss3 fc/* ~ 



where F is any arbitrary function of s and s'. Now the solution, which appears to 

 be most frequently useful, of the original equation is obtained by taking 



v = Ye-*", 

 where ^ is constant, and V is independent of x. In this case, 



and the equation for V is 



To identify the two forms, we must have 



TT /,. c.'\ - _1_ - ~ 



r 1 6, i ) j/ ^ ^ , "p . I ~~ 



7 ds os 2 ' \ 7 



= V - xp. V + ^V - . . . ; 



consequently 



V = F (*, 6-') 



_ /ji_\" 3 s "? f 



' 



for all values of n. All the conditions are satisfied in virtue of 



8'V 



3s3/ = iwV; 



and consequently the more general solution above obtained includes the less general 

 solution customarily used when the arbitrariness of F is made subject to the equation 



This equation is of LAPLACE'S linear form with equal invariants, which, moreover, 

 are constants ; hence the whole series* of derived invariants is unlimited in number, 

 and the number of derivatives of an arbitrary function of s and the number of deri- 

 vatives of an arbitrary function of s', that occur in the most general solution, are 

 unlimited in each case : that is, the solution is not expressible in finite terms. 



* DAKBOUX, ' Tlieorie Generale des Surfaces,' t. ii. 



