( 754 ) 



that oil tlie other hand the absolute vahie of ^ decreases in A for 

 increasing /•. For, if k becomes greater k' K decreases, but the deno- 

 minator k | en- w (he increases. 







Taking into consideration llie form just sketched of a curve OA 

 belonging to a definite value of /• we find that a second suchlike 

 curve belonging either to a larger value or to a smaller value of k 

 will certainly intersect the first curve somewhere. So as soon as a 

 cyclic minimal surface passes through two equal circles placed in 

 parallel planes we shall be able to bring a second cyclic minimal- 

 surface through these circles. 



We must now investigate when the two cyclic minimal surfaces 

 coincide, i. o. w. we must find the envelope of the curves (J A. 



If we put c-^^k", c'=^k'-, then the system of curves is given in 

 the equations 



u 



C ^^"' / 



B, = yc' en u I , ? = yc u en n ; 



J an* to 







we regard c as the parameter of the curve, </> = am u as the para- 

 meter determining a point on a given curve, so that the coordinates 

 (5 , 5) of a point of the envelope satisfy the condition 



D{e,\fi) 

 If here and in future we put for shortness' sake 



u u 



r dw r dw 



^ en' U' J an iv 







and we take into account that for constant ff = am u we have 



ÖU 1 



we find 



1 



= :; cn u Bill) , 



2l/c 



h' ds 



— = \/'e sn M {c Bill) - Q(u) ),—- = — V'csn u (c' B{u) + Q{u)) , 

 dip 0(f 



where Q{u) is given by the equations 



dn u C7i u 



Q{u) = u ~ E{n) , 



sn n 



