( 758 ) 



10 





?.'+-.";,= 1- 



In the table the values of (f^, 'ê„ and 5a calculated in this way 

 are added in the three last columns, to be compared. 



To conclude with we gi\e a computation of a part of a given 

 cyclic minimal surface with gi\'en parameters b and /■, situated 

 beween two equally large circles corresponding to the arguments -|- t/ 

 and — 11. 



The coordinates .r, y, :, of a point of the surface are again deter- 

 mined by the equations: 



b h , 



a r= bk' A (u) -\ cos «, 1/ znz sin «, z -r^ bk ii , 



en H ' en n 



out of which we can (ind for the line-element on the surface the 

 expression 



c/s* J^du — i en u da -\- i k' sin a du Pdu -f- i en u da — i k' sin a du 



= — - - ■ - - X :; 1 



A' en ti or u 



in which P is determined by the equation 



P- 1= (k' eos (( 4" sn n dn n)" -{- k" oi* u. 



We introduce for a an imaginary argument r. 



We substitute 



and we find 



tg ^ am v 

 ta \ a ^^ i 



sin a =::: 



COS a ■= 



tg h cim (w — K) 



i sn V sn {u — K) 

 en V — en {ti — K) 

 1 — en V en (m — K) 

 en V — en {u — K) 



and finally 



da dn v dn {u — K) 



sin a sn v sn (m —K) 



en" u dn V dti hi — K) 



p— —-~^, 



k' (en V — en {u — K) ) 



ds- d?i'' V dn" in — A') 



^ -;J—^{du—dv){du^dv). 



b" k'- (en v — en ( n — A') )" 



From this ensues that k -\- v and u — i' are the parameters of 

 the lines of length zero, so that r is the parameter of the greatest incline. 



According to the general properties of the minimal surfaces we have 

 for th« èuperlicial element dSi the expression 



