( V(i() ) 



^ r., .- ^' - K' 



~ =^ hL -\- t. ni u ' — .t?r uK'Q u) , 



4 k' "■ ' 



wliere ^ again i-epresents the ./'-cooi-dinate of the centre M of the 

 upper circle. 



If this centre M moves on the envelope BA of the diagram, then 

 u becomes equal to the critical argument u^ , Q (ic) equal to zero 

 and we have obtained the greatest possible minimal surface ii^ for 

 the given value of k. So 



Si E' — K' 



— = n.L + £„ en V, . 



We can now put the question where we have to put M on the 

 envelope BA, that is what value must be given to k for fi^, to 

 obtain the greatest possible value. To answer that question we sub- 

 stitute c = k- and <f^ = amit^\ then f2„ is a function of c, whilst 

 </u and ^Q are connected with c by means of the equations 



A' 



^ sn- IV 



"0 



By ditferentiation we lind 



dipg 1 



— - = — - sn- u^ dn », B (//,), 

 «c 2 



du^ 1 



dc 2c 



rff„ 1 



-— = -^ dn u^ B (?/„) {en n^ dn u^ -\- ?/„ c S7i- wj, 



dc 2Vc 



and tinally by means of these results 



d /i2„^ K'—E' 



en ?/„ dn »„ B (?/„) (en w„ </n v^ -|- ?/„ c .<t/i* mJ. 



As the right member of the last equation is always positive, ^0 

 always increases with c or with l\ The greatest possible surface 

 between the two circles is obtained by placing M in B, we have 

 then a part of the catenoid, of which half the height is equal to 

 CO/ 1? = 0.6627. 



3t 



Now K' = E' = 



2' 



^ = w„ = 1.1997. 

 2;r 



