SOLUTION OF A LINEAR PARTIAL DIFFERENTIAL EQUATION. 



553 



function of x and y, such that the resulting values of P, Q, R, p, q make (1) an 

 identity. 



We farther suppose, 1st, That the related values of (x,y,z) do not constitute a critical 

 point of any one of the functions P, Q, R. 



For example, if ~P={z— J(z — x—y)}/{z—2x—y}, the relations z = x + y and z = 2x + y are 

 excluded, because the first corresponds to a locus of branch-points of P, and the second to a locus of 

 infinity-points. 



2nd, That the related values of (x,y,z) do not cause all the three functions P, Q, R 

 to vanish. 



Any solution which has either of the two peculiarities just mentioned is classed as a 

 Singular Solution of the equation, and the following demonstration does not necessarily 

 apply to it. We shall speak of values of (x,y,z) which violate the two conditions above 

 (whether they constitute a solution of (1) or not) as Critical Values. 



The auxiliary system of Lagrange, viz., 



dx/¥ = dy/Q = dz/~R (3) 



has,* if we exclude critical values of x, y, z, the integral system 



u = a, v — b (4), 



where u and v are independent functions of (x,y,z), which may be taken to be single- 

 valued, continuous, and finite for all values of (x,y,z) which constitute an ordinary 

 solution of (1). If, for example, u became infinite for any such solution, we could 

 replace the integral u = a by l/u = a'. 



Now (4) gives us 



u x dx + Uydy + u z dz = ,\ /g\ 



v x dx + Vydy + v z dz — 3 

 whence, by (3), 



u x P+u-vQ + u z R = 0,} " ( 6 ) 



Since a and b do not occur in (6), and do occur separately in (4), the equations (6) 

 cannot be merely conditional, but must be identities. Therefore, since P, Q, R cannot all 

 vanish for any non-critical values of (x,y,z), we must have, for all such values, the 

 identities 



P/ 1 u v u 



Hence, 



y w z 

 Vy V z 



=Q/ 



u z u x 

 v z v x 



*/ 



Vx V v 



(7). 



Prop. I. If we neglect any factor which contains (x,y,z) alone, the equation (1) is 



equivalent to 



p q -1 =0 (8). 



U X Uy U z 



V X Vy V z 



* By Caucht's "Fundamental Theorem regarding the Existence of the Solution of a System of Differential 

 Equations." For a demonstration, see Madame Kowalewski, CrelMs Journal, Bd. lxxx. (1875). 



