DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 67 



hence 



2 / dw , dw\ 



v (P ~T + 2 T" ' 

 V (fo J rfy / 



no arbitrary function of u needing to be added, for it can be considered as accounted 

 for in the arbitrary function w. 



But now, regarding variations of u, we must have the equation 



do d? 



= n 

 du, du 



satisfied ; that is, the function w must satisfy the equation 



, d, I dw , dw\ . fiPw . d-w . 



7 T5 KW 



ay ) \(b: 2 dy- 



where A denotes 1 + xp' + y<2- As in the former example, this equation imposes 

 the limitation upon the arbitrariness of w; when its general integral is known, the 

 most general value of v can be deduced. 



40. It is an inference from the general theory that the quantity u, determined as 

 a function of x, y, z, by the equations 



z = u + xp(u] 



is an argument of the arbitrary functions that occur in the solution of the equation 

 V 3 v -f- K 2 v = 0. We therefore transform the variables from x, y, z, to x, y, u, where 

 u now is known ; and the result, obtained by analysis similar to that in 33, is 



d-o d-v -l 



Manifestly a function of u alone is not a solution of this equation ; but, on the 

 analogy of 35, we are led to consider what solutions (if any) of the form 



are possessed by the equation, 17 having its former value xp' + yq. When this value 

 is substituted, the equation takes the form 



where 

 Hence 



K 2 





