( -^' ) 



envelope of the curves OA luis al»i)ut the shape of a quadrant of 

 ellipse BA of wliieh iialf of the grea( axis OA = J and half of the 

 small axis OH = 0.6627. 



Moreover it is clear that the tangent to any curve k =: constant, in 

 the point P where the latter touches the envelope, is normal to the 

 tangent in the origin (^ drawn to this same curve. The preceding 

 calculations now lead to tiie conclusion that through the two equal 

 circles with radius li ■==. i placed parallel and symmetrically with 

 respect to the origin two cyclic minimal surfaces will pass, when the 

 centre /)/ (^,C) of the upper circle is situated inside the curve /).4 of the 

 diagram, that the two surfaces coincide when M has arrived on the 

 curve BA and that the circles cannot be connected by a minimal 

 surface when M falls outside the curve BA. 



If M lies inside the curve BA two curves ÜA pass through M. 

 One of these touches the envelope in B, a point on curve ÜA between 

 and M. So the argument u belonging to M is greater than the 

 critical argument u^ 'u\ B and so the minimal surface belonging to 

 it and extended between the circles M and AB would contain the two 

 circles along which this minimal surface is cut by a second minimal 

 surface with an intinitesimal slight difference. So this minimal surface 

 is unstable. For the second minimal surface laid through the circles 

 an argument n corresponds to M smaller than the critical argument 

 ^^(, ; this surface is therefore stable and can be realized in a proof 

 of Plateau. 



If two surfaces can be laid through the circles the most oblique 

 surface (the surface belonging to the smaller value of /.: and with 

 the greater value of the radius h of the mean section) is therefore 

 always stable, the other is unstable. 



It is worth mentioning that whilst here the quantities y „, ^o, ? 

 depend in rather an intricate way on k = sin 6, we can tind by 

 approximation out of simple formulae very accurate values for 

 these quantities. 



If we call the critital amplitude 56' 28' of the catenoid ^, we shall 



be able to assume with great accuracy the following relations : 



/ 4 , \ 

 cos (p^ z=. COS ^ sin- ó' j 1 -| cos- 6 I, 



\cos p J 



y cos ^ J 



from which ensues for the equation of the envelope BA 



