372 THE FORMS OF CELLS [ch. 



which corresponds a positive pressure P, suppHed by the surface- 

 tension as in the case of the sphere, but evidently of just half the 

 magnitude. 



In plane, sphere and cyUnder the two principal curvatures are 

 constant, separately and together; but in the unduloid the curva- 

 tures change from one point to another. At the middle of one of 

 the swollen "beads" or bubbles, the curvatures are both positive; 

 the expression (l/R + 1/^') is therefore positive, and it is also finite. 

 The film exercises (like the cyUnder) a positive pressure inwards, 

 to be compensated by an equivalent outward pressure from within. 

 Between two adjacent beads, at the middle of one of the narrow 

 necks, there is obviously a much stronger curvature in the trans- 

 verse direction; but the total pressure is unchanged, and we now 

 see that a negative curvature along the unduloid balances the 

 increased curvature in the transverse direction. The sum of the two 

 must remain positive as well as constant; therefore the convex or 

 positive curvature must always be greater than the concave or 

 negative curvature at the same point, and this is plainly the case 

 in our figure of the unduloid. 



The catenoid, in this respect a limiting case of the unduloid, has 

 its curvature in one direction equal and opposite to its curvature 

 in the other, this property holding good for all points of the surface ; 

 R = — R'; and the expression becomes 



{l/R -+- l/R') = (L/R - l/R) = 0. 



That is to say, the mean curvature is zero, and the catenoid, 

 like the plane itself, has no curvature, and exerts no pressure. 

 None of the other surfaces save these two share this remarkable 

 property; and it follows that we may have at times the plane and 

 the catenoid co-existing as parts of one and the same boundary 

 system, just as the cyHnder or the unduloid may be capped by 

 portions of spheres. It follows also that if we stretch a soap-film 

 between two rings, and so form an annular surface open at both ends, 

 that surface is a catenoid : the simplest case being when the rings are 

 parallel and normal to the axis of the figure*. 



* A topsail bellied out by the wind is not a catenoid surface, but in vertical 

 section it is everywhere a catenary curve; and Diirer shews beautiful catenary 

 curves in the wrinkles under an Old Man's eyes. A simple experiment is to invert 



