( 753 ) 



Farther we liave : 



11 u 



rd{tnw) re 



T=: k en a I <^ A; en u I - 



J dn w J 



''d (tn K^) ^ ,, fd {tn w) 



dn u 



SO that from 



follows : 



Jc sn u 



ClJl u 



-— = {k — .sn u dn u B) 



du en 71 



dS, 



"^ k' C7l u. 



du 



We conclude that for increasing u tlie variable ^ grows regularly 

 from to 1. So the curve O A is intersected but once by a line 

 § = constant. 



At the same time : 



d^ f sun \ 



— ■:=. k [en u — u .vi u dn ii) ■=. k en n \ 1 — n. j. 



dri \^ sn (u -j- K)J 



d^ 

 For small u we find — to be positive, it keeps on decreasing, becomes 

 dn 



one time zero and is then negative. So the variable ^ reaches 



somewhere a maximum and the curve OA is either not cut by 



a line S = constant or in two points. The form of the curve 



k = constant is therefore as is indicated schematically in the diagram. 



In order to l)e able to compare the curves belonging to different 



values of k we can determine the values which the differentialquotient 



— assumes in the points O and A. 

 di ^ 



(0) 



From this is apparent that in the value of — increases with k, 



ds, 



