Partial Differential Equation of the Second Order. 121 



the last of which is an equation of condition requiring to be sa- 

 tisfied by the coefficients of (1), in order that it may be capable 

 of being derived from a single equation of the form (I.). 



When the condition is satisfied, we have for the determination 

 of j b a partial differential equation of the first order involving 

 two independent variables ; and substituting this in the equation 



0=q-m 1 p- fizr-f bt 



we shall have the first integral of (1). 



II. The equations (5') hold always when the partial differen- 

 tial equation of the first order from which (1) is derivable con- 

 tains q ; but the results above derived from those equations fail 

 if/' (p) = 0, since in that case the foregoing elimination of A and 

 B from the equations (5) becomes illusory. 



In this case the equations (5') become 



B = S, =/J 



A/,+B/ a =-v 5=/ . 



So that if (1) be derivable from a single partial differential equa- 

 tion of the first order which does not contain^, such equation 

 must be of the form 



q=f(xyz), 



where we have simultaneously 



o=R, =/) 



0=T s=/ ./'(y) + S s=/ ./'W+(T, =/ ./+S, =/ . iJ )/V)+V s=/ . 



If we have 



V = Pp-fQ? + N.z + M, 



where R, S, T, P ; Q, M, N are functions of x and y only, the 

 above become 



= R, 



= T./'(y)+S./'(fl ? ) + (T./+S.^'(^)+Pi' + Qg + N2r + M; (12) 



and since /does not contain^?, the terms in this equation which 

 involve p must vanish separately. Hence we must have simul- 

 taneously 



= T./'(y) + S. /'(*) + T./(*)+Q./+N* + M/ (13) 

 0=S./'(*) + P (14) 



Integrating (13), we get 



p 

 f{xyz) = —gz+f tt (x,y); 



