{217}
If instead of a cylinder, which is curved only in one direction, we take a case where there are curvatures in two dimensions (as for instance a sphere), then the effects of these must be simply added to one another, and the resulting pressure p is equal to Tã₤ãã₤Rã₤+ã₤Tã₤ãã₤Rÿ£¢ãý or p =ã₤T(1ã₤ãã₤Rã₤+ã₤1ã₤ãã₤Rÿ£¢ãý)22.
And if in addition to the pressure p, which is due to surface tension, we have to take into account other pressures, pÿ£¢ãý, pÿ£¢ã°, etc., which are due to gravity or other forces, then we may say that the total pressure, P =ã₤pÿ£¢ãýã₤+ã₤pÿ£¢ã°ã₤+ã₤T(1ã₤ãã₤Rã₤+ã₤1ã₤ãã₤Rÿ£¢ãý). While in some cases, for instance in speaking of the shape of a birdãs egg, we shall have to take account of these extraneous pressures, in the present part of our subject we shall for the most part be able to neglect them.
Our equation is an equation of equiôÙlibôÙrium. The resistance to compression,ãthe pressure outwards,ãof our fluid mass, is a constant quantity (P); the pressure inwards, T(1ã₤ãã₤Rã₤+ã₤1ã₤ãã₤Rÿ£¢ãý), is also constant; and if (unlike the case of the mobile amoeba) the surface be homogeneous, so that T is everywhere equal, it follows that throughout the whole surface 1ã₤ãã₤Rã₤+ã₤1ã₤ãã₤Rÿ£¢ãý =ã₤C (a constant).
Now equiôÙlibôÙrium is attained after the surface contraction has done its utmost, that is to say when it has reduced the surface to the smallest possible area; and so we arrive, from the physical side, at the conclusion that a surface such that 1ã₤ãã₤Rã₤+ã₤1ã₤ãã₤Rÿ£¢ãý =ã₤C, in other words a surface which has the same mean curvature at all points, is equivalent to a surface of minimal area: and to the same conclusion we may also arrive through purely analytical mathematics. It is obvious that the plane and the sphere are two examples of such surfaces, for in both cases the radius of curvature is everywhere constant, being equal to infinity in the case of the plane, and to some definite magnitude in the case of the sphere.
From the fact that we may extend a soap-film across a ring of wire however fantastically the latter may be bent, we realise that there is no limit to the number of surfaces of minimal area which may be constructed or may be imagined; and while some of these are very complicated indeed, some, for instance a spiral helicoid screw, are relatively very simple. But if we limit ourselves to {218} surfaces of revolution (that is to say, to surfaces symmetrical about an axis), we find, as Plateau was the first to shew, that those which meet the case are very few in number. They are six in all, namely the plane, the sphere, the cylinder, the catenoid, the unduloid, and a curious surface which Plateau called the nodoid.
These several surfaces are all closely related, and the passage from one to another is generally easy. Their mathôÙeôÙmatôÙiôÙcal interrelation is