( 755 ) 



K 



r dw 



K-E- -^, 

 ^ an 2v 



1 



z=. — — (n — C7I* n A(n) — d7i* u B{u)), 



1 



— A{n) -\- k' B{u) — 



sn u en ?/ dn u 

 From this ensues 



777 ^ = — ;r-T-, <^n u sn n B{u) Q{u) , 



T>{c,(f) 2 l/cc 



and so the points of the envelope of the curves ()A are determined 

 by the equations 



K 



Q(u)=K-E - -^ = ^). 



As when c is given, the tirst member of the equation increases 

 regularly from — oo for ?/ =: to K — E for u = K, the equation 

 Q{ii)^=0 admits of one solution u^. By differentiating we find 



du,^ 1 r _ Vdn^u^ 



1 r Vi 



dc 2c ^ I dn^ 







i. e. a negative value ; therefore the greater c is, the smaller is the 

 argument u^, which I call the critical argument. This argument 

 moves finally between rather narrow limits. For c = we find 



K=z E=— and so also u^ =: - =1.5708. For c = 1 we find 



„ , , ^ ^ ^ dnu cnu 1 Chu 



Q (n) — u - E {u) — — u = u — ---— - . 



sn n xn ii aSA u 



So the critical argument «„ satisfies the equation 



Ch u,. 



Sh u^ 

 From this ensues 



w„ = 1.1997, 



(p„ = am ?/.„ = 56°.28', 



cot (fi^ := u„ en u^ = 0.6627. 



1) G. JuGA. (Ueber die Constantenbestimmung be! einer cyklischen Mlnimaltlache, 

 Math. Ann. Bd. 52) gives this equation in the form 



cnu dnu -j- (E {u) — //) s?iu = 0. 



