Of all the surfaces which we have been describing, the sphere is the only one which can enclose space; the others can only help to do so, in combination with one another or with the sphere itself. Thus we have seen that, in normal equilibrium, the cylindrical vesicle is closed at either end by a portion of a sphere, and so on. Moreover the sphere is not only the only one of our figures which can enclose a finite space; it is also, of all possible figures, that which encloses the greatest volume with the least area of surface; it is strictly and absolutely the surface of minimal area, and it is therefore the form which will be naturally assumed by a unicellular organism (just as by a raindrop), when it is practically homogeneous and when, like Orbulina floating in the ocean, its surroundings are likewise practically homogeneous and symmetrical. It is only relatively speaking that all the rest are surfaces minimae areae; they are so, that is to say, under the given conditions, which involve various forms of pressure or restraint. Such restraints are imposed, for instance, by the pipes or annuli with the help of which we draw out our cylindrical or unduloid oil-globule or soap-bubble; and in the case of the organic cell, similar restraints are constantly supplied by solidification, partial or complete, local or general, of the cell-wall.
Before we pass to biological illustrations of our surface-tension figures, we have still another preliminary matter to deal with. We have seen from our description of two of Plateau’s classical experiments, that at some particular point one type of surface gives place to another; and again, we know that, when we draw out our soap-bubble into and then beyond a cylinder, there comes a certain definite point at which our bubble breaks in two, and leaves us with two bubbles of which each is a sphere, or a portion of a sphere. In short there are certain definite limits to the dimensions of our figures, within which limits equilibrium is stable but at which it becomes unstable, and above which it {226} breaks down. Moreover in our composite surfaces, when the cylinder for instance is capped by two spherical cups or lenticular discs, there is a well-defined ratio which regulates their respective curvatures, and therefore their respective dimensions. These two matters we may deal with together.
Let us imagine a liquid drop which by appropriate conditions has been made to assume the form of a cylinder; we have already seen that its ends will be terminated by portions of spheres. Since one and the same liquid film covers the sides and ends of the drop (or since one and the same delicate membrane encloses the sides and ends of the cell), we assume the surface-tension (T) to be everywhere identical; and it follows, since the internal fluid-pressure is also everywhere identical, that the expression (1 ⁄ R + 1 ⁄ R′) for the cylinder is equal to the corresponding expression, which we may call (1 ⁄ r + 1 ⁄ r′), in the case of the terminal spheres. But in the cylinder 1 ⁄ R′ = 0, and in the sphere 1 ⁄ r = 1 ⁄ r′. Therefore our relation of equality becomes 1 ⁄ R = 2 ⁄ r, or r = 2 R; that is to say, the sphere in question has just twice the radius of the cylinder of which it forms a cap.
Fig. 67.
And if Ob, the radius of the sphere, be equal to twice the radius (Oa) of the cylinder, it follows that the angle aOb is an angle of 60°, and bOc is also an angle of 60°; that is to say, the arc bc is equal to (1⁄3) π. In other words, the spherical disc which (under the given conditions) caps our cylinder, is not a portion taken at haphazard, but is neither more nor less than that portion of a sphere which is subtended by a cone of 60°. Moreover, it is plain that the height of the spherical cap, de,
= Ob − ab = R (2 − √3) = 0·27 R,
where R is the radius of our cylinder, or one-half the radius of our spherical cap: in other words the normal height of the spherical cap over the end of the cylindrical cell is just a very little more than one-eighth of the diameter of the cylinder, or of the radius of the {227} sphere. And these are the proportions which we recognise, under normal circumstances, in such a case as the cylindrical cell of Spirogyra where its free end is capped by a portion of a sphere.
Among the many important theoretical discoveries which we owe to Plateau, one to which we have just referred is of peculiar importance: namely that, with the exception of the sphere and the plane, the surfaces with which we have been dealing are only in complete equilibrium within certain dimensional limits, or in other words, have a certain definite limit of stability; only the plane and the sphere, or any portions of a sphere, are perfectly stable, because they are perfectly symmetrical, figures. For experimental demonstration, the case of the cylinder is the simplest. If we produce a liquid film having the form of a cylinder, either by