DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 01 



If2>( u ) an d q(u) denote any functions ofu satisfying the equation 



p* + ,/ + i = o, 

 and ifube defined by the equation 



z = u + xp (u) + yq (w), 

 then 



where <f> and t// are arbitrary functions, satisfies LAPLACE'S equation 



V 3 w = 0. 



36. The solution just obtained can, like the solution of 33, be connected with 

 the general solution. Owing to the linear form of all the equations and of the 

 expression for u, it will be sufficient, in the first instance, to take the part 



. 



- 



1 + r, " 1 + 



say ; because the term (f> (u) in v is, in effect, identified by the preceding case. We 



thus must have 



d'lv dw ir 



the most general solution of which is 



= nh;J + H (* *) 



where, so far as concerns this relation, H is any arbitrary function of both its 

 arguments. We have 



dhu _2 __ j/ 2a y> ;2 -f , 3 8 3 H 



~d# = ~ (1 + 17)* ~p~ "~ (1 + nT P ~ P " &)* ' 



_J __ 2lt, 2x PV . O v 



~ ~ 2 r r p v 



/a 

 ' q 



and therefore, after some simple reductions on substituting in the expressions of 31, 

 we have 



