In the cylinder, one radius of curvature has the finite and positive value R; but the other is infinite. Our formula becomes T ⁄ R, to which corresponds a positive pressure P, supplied by the surface-tension as in the case of the sphere, but evidently of just half the magnitude developed in the latter case for a given value of the radius R.
The catenoid has the remarkable property that its curvature in one direction is precisely equal and opposite to its curvature in the other, this property holding good for all points of the surface. That is to say, R = −R′; and the expression becomes
(1 ⁄ R + 1 ⁄ R′) = (1 ⁄ R − 1 ⁄ R) = 0;
in other words, the surface, as in the case of the plane, has no {222} curvature, and exercises no pressure. There are no other surfaces, save these two, which share this remarkable property; and it follows, as a simple corollary, that we may expect at times to have the catenoid and the plane coexisting, as parts of one and the same boundary system; just as, in a cylindrical drop or cell, the cylinder is capped by portions of spheres, such that the cylindrical and spherical portions of the wall exert equal positive pressures.
In the unduloid, unlike the four surfaces which we have just been considering, it is obvious that the curvatures change from one point to another. At the middle of one of the swollen portions, or “beads,” the two curvatures are both positive; the expression (1 ⁄ R + 1 ⁄ R′) is therefore positive, and it is also finite. The film, accordingly, exercises a positive tension inwards, which must be compensated by a finite and positive outward pressure P. At the middle of one of the narrow necks, between two adjacent beads, there is obviously, in the transverse direction, a much stronger curvature than in the former case, and the curvature which balances it is now a negative one. But the sum of the two must remain positive, as well as constant; and we therefore see that the convex or positive curvature must always be greater than the concave or negative curvature at the same point. This is plainly the case in our figure of the unduloid.
The nodoid is, like the unduloid, a continuous curve which keeps altering its curvature as it alters its distance from the axis; but in this case the resultant pressure inwards is negative instead of positive. But this curve is a complicated one, and a full discussion of it would carry us beyond our scope.
Fig. 63.
In one of Plateau’s experiments, a bubble of oil (protected from gravity by the specific gravity of the surrounding fluid being identical with its own) is balanced between two annuli. It may then be brought to assume the form of Fig. [63], that is to say the form of a cylinder with spherical ends; and there is then everywhere, owing to the convexity of the surface film, a pressure inwards upon the fluid contents of the bubble. If the surrounding liquid be ever so little heavier or lighter than that which constitutes the drop, then the conditions of equilibrium will be accordingly {223} modified, and the cylindrical drop will assume the form of an unduloid (Fig. [64] A, B), with its dilated portion below or above,
