DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 51 



Since 



dv dv dz 



= I -J- np, = n - 

 ax du du 



are satisfied by the integral relations, we have 



d d f dz\ 



(( + np) = (n - - , 

 du ^ dx\ du] 



and therefore 



( ]L _i_ dn d* 

 du " dn dx du 



the terms in -~- cancelling ; consequently 



dl, fdn \ dz dz 



= PC = 7 - 

 du \(lx J du J du 



Similarly, from 



dv dv dz 



= m + nq, - = n 

 dy dn du 



we find 



dm . dz 



du ^ du 



the quantities g and f which occur here being those which formally occur in the 

 derivatives of I, m, n, with regard to x and y. 

 Again, we have 



- 3?; . dl , dl . dl 



= d =- =. dl = dx + -7- dii + -7- 



3,r rf. rf y du 



whence 



^ 



du 



and therefore, comparing with the former equations, we have 



3% , d 2 v (?v 



= a^' Al = cte3y' ^" 



Similarly we find 



7 _ 3^ ,._ ^_ _ 



^ ^~* ^ o, ^/' -^ ^ O~ J ^ 



3/ o^ 



Now, when we take the combination 



H 2 



