Partial Differential Equation of the Secona Order. 233 



Since /contains U and F does not, (25) gives us 



= RFV) + SF%) + 11.F.F(^)+PF + N.? + M, . (27) 

 = S.F(*)+Q. 



The last gives us 



and 





Substituting these values in (27) and putting the coefficient of z 

 in the result =0, we get 



= RF„' (») + SF.'fr) + (p - Q §) F„ +M, . . (28) 



«£©+'*®+(43>it* 



the last of which equations was shown in my previous paper to 

 be the condition to be satisfied by the coefficients, in order that 

 the given equation may be derivable from a single equation of 

 the first order into which p does not enter ; while (28) affords 

 the means of ascertaining such single equation. 



F being determined, (26) gives us the value off which will 

 contain an arbitrary function of y. 



If F be supposed to contain TJ-u we must go through much 

 the same process as is above adopted in the general case when 

 the integral takes the form of a series of derivatives of <£; the 

 details I here omit. 



A reference to (6) will show that the results derived from 

 those equations in the earlier part of this paper fail when R. or 

 T vanishes. I believe that these cases of failure, however, will 

 be found to resolve themselves into that which has already been 

 discussed. 



With regard to the general tenor of the results of this and 

 of my previous paper, I offer the following remarks. 



A partial differential equation of the second order between 

 xyz may always be replaced by two partial differential equations 

 of the first order between oc, y, z,p, q (see 5a). That such a pair of 

 equations should be derivable from a single equation between 

 oc } y,z,p,q is obviously what cannot generally occur; i.e. its occur- 



