( 553 ) 



§ 7. We can now put the question whether the equations {VII) 

 can be solved bj putting u equal to /{v), wliere for the present ƒ 

 is arbitrary. 



From ( VII ) can be deduced 



didyi dSdri 



dv dv du du 



For u=/ {v) this leads to 



du dv 



d^u dv dv , d'v 



So: 



d^dri d^ dt] ögöï^ 



d'u „ dv dv , d^v 



or 



öl • ö7j -^ ^''^ ag ' dri 



öu öu v — u d^v 

 Then, according to (Vil), aj ' aV^ ^ "F * öP^* 



So: 



r(^')+/'(^) + — ^/"(^) = o. 



One integral of this is sufficient to recognize the nature of the 

 surfaces found. We find that satisfies 

 f{v)=-v^). 



1) Prof. W.Kapteyn was so kind as to draw my attention to the following general 

 solution of the differential equation. 

 Put 



then 



2 dv^ "^ \^dvj "*" dv 



Now put 



so 



so that 



y = v -{- w, 



dy dio d^y d^w 



- = l H , — == , 



dv dv dv^ dv^ 



38* 



