382 THE FORMS OF CELLS [ch. 



site of the freshwater polype, we have a circular disc bounded 

 (apparently) by two parallel rings of cilia, with a pulley-like groove 



between. The groove looks very like 

 that catenoid surface which we have 

 produced from two parallel and 

 opposite annuli; and the fact that 

 the lower surface of the little creature 

 is practically plane, where it creeps 

 ^.^^^^^^^^^ over the smooth body of the Hydra, 

 i\ '\\\ vvjN ^ looks like confirming the catenoid 

 Fig. 112. Triclwdiruz pediculus. analogy. But the upper surface of 



the infusorian, with its ciliated 

 "gullet," gives no assurance of a zero pressure; and we must 

 take it that the equatorial groove of Trichodina resembles, or 

 approaches, but is not mathematically identical with, a catenoid 

 surface. 



While those figures of equilibrium which are also surfaces of 

 revolution are only six in number, there is an infinite number of 

 other figures of equilibrium, that is to say of surfaces of constant 

 mean curvature, which are jiot surfaces of revolution; and it can 

 be shewn mathematically that any given contour can be occupied 

 by a finite portion of some one such surface, in stable equilibrium. 

 The experimental verification of this theorem lies in the simple fact 

 (already noted) that however we bend a wire into a closed curve, 

 plane or not plane, we may always fill the entire area with a con- 

 tinuous film. No more interesting problem has ever been pro- 

 pounded to mathematicians as the outcome of experiment than the 

 general problem so to describe a minimal surface passing through 

 a closed contour; and no complete solution, no general method of 

 approach, has yet been discovered*. 



Of the regular figures of equihbrium, or surfaces of constant mean 

 curvature, apart from the surfaces of revolution which we have 

 discussed, the helicoid spiral is the most interesting to the biologist. 



* Partial solutions, closely connected with recent developments of mathematical 

 analysis, are due to Riemann, Weierstrass and Schartz. Cf. (int. al.) G. Darboux, 

 Theorie des surfaces, 1914, pp. 490-601; T. Bonneson, Problemes des i^operimetres 

 et des iaipipJianes, Paris, 1929; Hilbert's AnschauUche Geometrie, 1932, p. 237 seq.; 

 a good account also in G. A. Bliss's Calculus of Variations, Chicago, 1925. See also 

 (int. al.) Tibor Rado, Mathem. Ztschr. xxxir, 1930; -Jesse Douglas, Amer. Math. 

 Journ. XXXIII, 1931, Journ. Math. Phys. xv, 1936. 



