9 



Partial Differential Equation of the Second Order. 227 

 where 



"Hr-^u-i ^ 



Substituting the above value of f a in (13 1 ), and eliminating 

 U.! from the result by means of (16), we shall get/ of the form 



/ V =C*+D, 

 where CD are functions of xyco only. But as/ does not con- 

 tain U_ 1 , neither can it contain oy ; hence we must have 



dco V} 

 and therefore dC __ dl> __ 



dco dco 

 It will be found that 



where h involves x and y only, and 



1 f da da — - \ . 



Hence dC _ g ,„ _ 



Therefore either we must have g = 0, which, as has already 

 been pointed out, is the criterion of the given equations being 

 derivable from a single partial differential equation of the first 

 order; or we must have /,"(a))=0, from which it follows that 

 /j must be linear with respect to co, so that we may take 



= A+(, i + f / ft )*-/ m .U_ I) .... (17) 



where £ = — .-—, and f a and A are functions of x and y which 

 fp fy Jfil 



do not involve U_! or higher derivatives of </>, but may involve 

 lower derivatives of <j>. 



(17) gives us, observing that f a =f a (xyzV _ 1 ) f 



Q2 



