Tension of Liquid Films. 193 



very important conclusion follows, which has already been no- 

 ticed by M. Lamarle* ; for, imagine that through a tangent to 

 the curve a section is made normal to the surface. According 

 to Meusnier's theorem, any oblique section whatever passing 

 through the same tangent will have a radius of curvature equal 

 to the projection, on the plane of this section, of the radius of cur- 

 vature of the normal section. This being granted, if this theorem 

 be applied to the section passing through the osculating plane 

 of the curve, it is found that the normal section must necessarily 

 have an infinite radius of curvature. 



It follows thence that the curves formed by the flexible thread 

 on liquid films are precisely those which M. Dupin has called 

 asymptotic linesj-, or those which Mr. Michael Roberts has called 

 generating lines%. There arises then the interesting question 

 which forms the subject of the present research — that of ascer- 

 taining whether the properties of the lines of equilibrium of ten- 

 sion can be reconciled with the nature of the minimum surface 

 operated upon. 



We know that the surfaces of mean zero curvature may be 

 represented, according to Monge, by the three following equa- 

 tions : — 



x — u + v, -n 



y = $(u)+^r{v); L g (i) 



z= \S -\\^Vl+{$\u)¥du+$</l + [$\vyfdv} J 



where <j>(u), ^(v) are any functions of the arbitrary quantities u 

 and v. On the other hand, we have for the general equation of 

 the asymptotic lines traced on minimum surfaces (sec the memoir 

 by Mr. Michael Roberts) 



ft and i/r f , <p n and ijr" designate respectively the first and second 

 derivatives of the functions (p(u) and yfr(v). The question is to 

 see in each particular case if the curves represented by equation 

 (2) have or have not the same curvature at all points. In the 

 first case the thread might be placed on the laminar surface 

 without producing any deformation ; in the second case, on the 

 contrary, the thread could not maintain itself on the film, or if 

 it did, the form of the surface would necessarily be altered. 



* Compare M. Lamarle's report on my first research (Bull, de V Acad. 

 Roy. de Belgique, ser. 2. vol. xxii. p. 272). 



f Developpements de Geometrie, p. 189. 



% " On the Surfaces whose radii of curvature are equal but in opposite 

 directions," Journal de Lioaville, vol. xi. p. 302. 



