( 750 ) 



Mathematics. — "On the ci/c/ic iii'uwi'al surface". By Prof. J. C. 

 Kluyvek. 



(Communicated in the meeting of January 25, 1908). 



EiNNKPEK (Zeitsclir Math. I'lijis. 14) pointed to the existence of a 

 minimal surface containing a system of circles lying in i)arallel planes, 

 with centres sitnated on a [tlane cnrve. Let ns snppose that this 

 curve passes through the origin of the rectangular coordinates, that 

 it is situated in the A'Z-plane and that the varial)le circle with the 

 centre (£, 0, $) and the radius A*, generatijig the surtace, lies always 

 in a plane })arallel to the A'}'-plane. 



The rectangular coordinates ,c, //, : of a point of the surface are 

 given by the equations 



.6' := § -j- Ji COf< (( , ,?/ = li )iln n , ^ = ?, 



so that they are expressed in the two parameters a and ^. We find 

 that the dilferential equation of the minimal surfaces is satisfied when 



R' if.R CO*' a 4- RR") — R (I + è" + H'' + H' -}- - s' H' cos u) = 0, 

 in which equation the dashes denote the dilferentiations with regard 

 to ?. 



The equation breaks up into 



and into 



RR" = 1 + é" + 7^". 



The first equation furnishes 



where .1 denotes a positive constant and h the minimum value of /?. 

 The second equation now passes into 



(/ rR'\ 1 A'R' 



d^ yRj , R' b' 



and the integration furnishes 



1 / .IVt!- 



so that tinally we can express i en s in H l)y means of elliptic integrals. 

 We find 



R R 



A r dR ^ r RdR 



BJ. / / A'K'\ J { / ( ^■B?\ 



