222 THE FOEMS OF CELLS [ch. 



curvature, and exercises no pressure. There are no other surfaces, 

 save these two, which share this remarkable property ; and it 

 follows, as a simple corollary, that we may expect at times to have 

 the catenoid and the plane coexisting, as parts of one and the 

 same boundary system; just as, in a cylindrical drop or cell, the 

 cylinder is capped by portions of spheres, such that the cylindrical 

 and spherical portions of the wall exert equal positive pressures. 

 In the unduloid, unlike the four surfaces which we have just 

 been considering, it is obvious that the curvatures change from 

 one point to another. At the middle of one of the swollen 

 portions, or "beads," the two curvatures are both positive; the 

 expression {IjR + 1/-R') is therefore positive, and it is also finite. 

 The film, accordingly, exercises a positive tension inwards, which 

 must be compensated by a finite and positive outward pressure 

 P. At the middle of one of the narrow necks, between two 

 adjacent beads, there is obviously, in the transverse direction, 

 a much stronger curvature than in the former case, and the curva- 

 ture which balances it is now a negative one. But the sum of the 

 two must remain positive, as well as constant ; and we therefore 

 see that the convex or positive curvature must always be greater 

 than the concave or negative curvature at the same point. This 

 is plainly the case in our figure of the unduloid. 



The nodoid is, like the unduloid, a continuous curve which 

 keeps altering its curvature as it alters its distance from the axis ; 

 but in this case the resultant pressure inwards is negative instead 

 of positive. But this curve is a complicated one, and a full 

 discussion of it would carry us beyond our scope. 



In one of Plateau's experiments, a bubble of oil (protected from 

 gravity by the specific gravity of the surrounding fluid being 

 identical with its own) is balanced between two 

 annuli. It may then be brought to assume the form 

 of Fig. 63, that is to say the form of a cylinder with 

 spherical ends ; and there is then everywhere, owing 

 to the convexity of the surface filjn, a pressure 

 inwards upon the fluid contents of the bubble. If 

 the surrounding liquid be ever so little heavier or 

 lighter than that which constitutes the drop, then 

 the conditions of equihbrium will be accordingly 



