Equations of the First Order . 341 



one of the equations of condition (xii) be not satisfied identically, it will 

 constitute a new partial differential equation to be combined with the 

 given system (xi). Let such combination be presented in the form 



/i =Pi +*\ (x v . . . cc n+r , p n+2 , . . .p n+r )=0, -J 



: [ (xiv) 



Treating this system in the same way as we have just done (xi), we either 

 get a solution of the form 



z=<j>(x v . .. ar n+ri a v . ..ctr-^ + b, 



or fall upon a system of n+2 equations analogous to (xi) and (xiv). Pro- 

 ceeding in this way we must finally arrive at a solution of the form 



z=$(x v oe 2 , . ..cc n+r , a v a 2 , . .. a s ) + b, 

 where n is less than r, or else we shall have the system 

 Pi +F, (x v # aJ . . . a?„ +r ) =0, 



Pn+r-\-^n+ r (^\, ®& • •• #»+r)=0 J 



and if the |(n-r-r)(n+r— 1) conditions 



dFj _dFj _~ 



are satisfied, we have a solution 



z=(j)(.v v cv 2 , ,...x n+r ) + b; 

 but if these conditions are not satisfied, then there is no solution. 

 Example 1. Required the integral of the simultaneous equations 



, dz , , dz 



dx l cuv n 



Let 



(j>(z, x\, . ..x n ) — 



be any integral, and let 



d<f) dfi 



dz' dx r ' 



then 



f 2 =\p x -\-b 2 p 2 + • • • + bfp n + ep = 0, 



and the condition [/ 1 ,/ 2 ]=0 is satisfied. Also it is easily seen that the 

 functions / 3 =p 1 ,/ 4 = J p 2 , . ../ w +i =prt_i satisfy the requisite conditions. 

 Hence we take 



Pi = a v Pz = a » •-Pn = a n , 



z 



