194 M. G. Van der Mensbrugghe on the 



1. Plane Surface. 



If we put (p" = Q } ilr" — Q } we shall find, after having eliminated 

 u and v in the equations (I), a result of the form 

 z — Ax -f B?/ -f C ; 



consequently the minimum surface is then a plane. As the cur- 

 vature of the lines formed by the thread must be constant, this 

 latter will always assume a circular form. This I verified by 

 numerous experiments in my first investigation. 



2. Skew Helicoid. 



Let us determine, in the second place, the functions </> and yjr. 

 Taking _ 



</>(V)= /-WI+m 2 , 



equations (1) give then, as we know, 



x 

 z= — arc tan -, 



y 



and represent therefore a skew helicoid ; equation (2) becomes 

 du dv 



In order more easily to integrate this differential equation, 

 let us put 



u — s/ — 1 sin X, v — sf — 1 sin p, 

 we shall thus have 



x=. V — 1 -jsin\+ sin/x}, 

 yz= \/ — 1 |cos>v+ cos^j-, 



*=-(*. + /*), 

 d\~ +dfju. 



If we take the sign — , we have A 4- ft= ^z — constant. We 

 obtain thus all the rectilinear generating lines of the helicoid. 

 If we take, on the contrary, the sign +, we find 



\-/x = C; ^H/=-4cos 2 /^^j=-4cos C2 ^. 



Making C = tt-\-2u */ — 1, we find 



a formula which represents, besides the axis of the helicoid, the 

 totality of all the helices of the same pitch which can be traced 

 on the surface; we know, on the other hand, that the helix has 

 a constant curvature. • Hence it follows that the lines of equiii- 



