( 751 ) 



Here an elliptic argument can be introduced. We put 



6 



R = 



en u 



sin 6 :==■ 



and we And 



[/I -\- A' 



u 



rdw 



^ C7l W 



U\ allowinj; u, lo vary from — K to + K the centre J/ with rhe 

 coordinates i:, ? in the A/f-plane describes completely the locus of 

 the centres and the equation 



b 



R = 



cnu 



iiKhcates how the radius of the circle changes during the motion. 



We notice that the minimal surface depends on two constants b 

 and k, that the smallest circle (?^ = 0) is found in the .Yi'-plane, 

 that with respect to the origin there is symmetry, and that for u = K, 

 ? =r bkK the radius R has become infinite whilst at the same time 

 the centre AI is at infinite distance. 



As however 



Lim ($ 



-R) 



b Lim 

 u = K 



'ƒ 



dio 



1 ■ 



en u 



~ {k-- K-E) 



and s — R retains therefore a finite value the surface contains two 

 right lines 



z = ±bkK, 



b 



x—±-{k''K—E). 

 k 



For k = 1 the eliptic integrals degenerate. We have 

 1 = 0, ^z=z bu, R = bCh /<, 

 and the surface has passed into a catenoid. The smaller k is, the 

 more the surface deviates from the catenoid and the more obli{pie 

 it becomes. For, we find for the coefttcient of direction of the tangent 

 to the locus of the centres M : 



d^ k cn^ 11 



and the greatest value of this coefficient k : k' , which is arrived at in 

 the origiji, tends to zero when k tends to zero. The surface is then 

 altogether in the A'F-plane. 



51* 



