While the figures of equilibrium which are at the same time surfaces of revolution are only six in number, there is an infinite {230} 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 indistinguishable 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 Spirochætes, _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.