DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 59 



so that 



pp + qq = o, 



say 



q -p 



and so 



d-w _ J_ cPw _ J^ d\o 



' ~ ~ 



q- dx? ' py dxdy 



Consequently 



Now we should have 



dy rfe . 



= n = (1 + rj) n, 



(lu du ' ' 



that is, 



from which the form of G is given by 



G = A (u) + >?B (u) - |/' (u) [(1 + ,) (log (I + ,) - I}], 



A and B being arbitrary functions. 



The form of G is, however, not so important for the present purpose as is the value 

 of a'Gar 2 . We deduce 



1/93 _ _ 



O V r\ n 



1 +*?' 



and consequently 



? M /' (it) 



p q 1 1 + a?/ + ?/<?' 



being the proper values of I, m, n, as given by 



v=f(u) 



z = u -\- xp -\- yq 



The particular solution is thus seen to be included in the general solution defined 

 by the equations of 31. 



35. It is worth inquiring whether, in the notation of the preceding articles, there 

 is any solution of the potential equation which is a function of u and rj alone, say 



v = F (u, 17), 

 I 2 



