230 THE FOEMS OF CELLS [ch. 



number of figures of equilibrium, that is to say of surfaces of 

 constant mean curvature, which are not 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 may bend a wire 

 into a closed curve, plane or not plane, we may always, under 

 appropriate precautions, fill the entire area with an unbroken 

 film. 



Of the regular figures of equilibrium, that is to say 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. This is a helicoid generated by a straight line 

 perpendicular to an axis, about which it turns at a uniform rate 

 while at the same time it slides, also uniformly, along this same 

 axis. At any point in this surface, the curvatures are equal and 

 of opposite sign, and the sum of the curvatures is accordingly nil. 

 Among what are called "ruled surfaces" (which we may describe 

 as surfaces capable of being defined by a system of stretched 

 strings), the plane and the helicoid are the only two whose mean 

 curvature is null, while the cylinder is the only one whose curvature 

 is finite and constant. As this simplest of helicoids corresponds, 

 in three dimensions, to what in two dimensions is merely a plane 

 (the latter being generated by the rotation of a straight line about 

 an axis without the superadded gliding motion which generates 

 the helicoid), so there are other and much more complicated 

 helicoids which correspond to the sphere, the unduloid and the 

 rest of our figures of revolution, the generating planes of these 

 latter being supposed to wind spirally about an axis. In the case 

 of the cylinder it is obvious that the resulting figure is indistinguish- 

 able from the cylinder itself. In the case of the unduloid we 

 obtain a grooved spiral, such as we may meet with in nature (for 

 instance in Spirochsetes, Bodo gracilis, etc.), and which accordingly 

 it is of interest to us to be able to recognise as a surface of minimal 

 area or constant curvature. 



The foregoing considerations deal with a small part only 

 of the theory of surface tension, or of capillarity: with that 

 part, namely, which relates to the forms of surface which are 



