DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 55 



these constants occurring in association with u constant. We may, therefore, assume 



2 = u + xp (u) + yq (u) , 



where p and q are arbitrary functions of u, subject solely to the condition 



P* + </ +1 = 0. 



Because the differentiations with regard to x and to y are effected on the hypothesis 

 that u is constant, the other equation can be taken in the form 



d d 



- (I np) + -- (in no) = 0. 

 ax x dy v 



so that a function of x and y (and possibly also involving u) exists such that 



I np = i "i nq = 

 dy dx 



Moreover, we have 



dv dv 



t + np = > r/i + nq = > 

 dx dy 



consequently 



dv d% do r' 



+ 

 dy dx 



From the last two equations, it follows that 



. J dx 

 and thence that 



Consequently a function w of x and // (and possibly also involving u) exists such 



that 



. dw 



pv + tf = - , 



whence 



duo dw 



