( 557 ) 



dv dv 



By substitution of v — u, — and ^ we thus find 



'' OS ov 



df dv / \^f ^'' 



[V — U) 



1 = 



(V — U)ri--.— (t? — W) ^ . 



d'edn 

 dy dv 



"" d'v df 

 d^dy] ' dyj 

 d'v dy 



dè^^l d^dri 



~d7~ ~ ~ ~df ' du '~~~df 



dri dti d§ d| 



After integration we find : 



dv df , du df 



dv du 



Joining these equations to the values of — and — , we find : 



o§ di] 



dv du 1 , „ , du dv 1 



a, • 5, = - ¥ <" - "' ^' <"> ^"-^ el ■ el = ¥ '" - ">' ^- «'■ 



These equations must be regarded as the intermediate integrals; 

 thej contain the arbitrary functions F, and F^^ , and it is easy to 

 prove that by ditferentiation they lead back to the two equations 

 (VII) of order two. 



It goes almost without saying that F^ and F^ appearing here are 

 closely connected to f\ and ƒ, appearing in (VIII). 



From the equations just found follows : 



dv dv dw du 1 



or 



FAri)-Fr{i) 



E\ 



du dv du du 

 dr]' d§' ö>j ' d§ 



{v — uy 

 If we compare this to (X), then : 



-4i^,(7j)./^,(ê) = ^Vi(^)A(^). 



The first integrals found satisfy therefore all the conditions entirely. 

 We have transformed our original coordinates in such a way that 



