The physical transition of one of these surfaces into another can be experimentally illustrated by means of soap-bubbles, or better still, after the method of Plateau, by means of a large globule of oil, supported when necessary by wire rings, within a fluid of specific gravity equal to its own.
To prepare a mixture of alcohol and water of a density precisely equal to that of the oil-globule is a troublesome matter, and a method devised by Mr C. R. Darling is a great improvement on Plateau’s[287]. Mr Darling uses the oily liquid orthotoluidene, which does not mix with water, has a beautiful and conspicuous red colour, and has precisely the same density as water when both are kept at a temperature of 24° C. We have therefore only to run the liquid into water at this temperature in order to produce beautifully spherical drops of any required size: and by adding {220} a little salt to the lower layers of water, the drop may be made to float or rest upon the denser liquid.
We have already seen that the soap-bubble, spherical to begin with, is transformed into a plane when we relieve its internal pressure and let the film shrink back upon the orifice of the pipe. If we blow a small bubble and then catch it up on a second pipe, so that it stretches between, we may gradually draw the two pipes apart, with the result that the spheroidal surface will be gradually flattened in a longitudinal direction, and the bubble will be transformed into a cylinder. But if we draw the pipes yet farther apart, the cylinder will narrow in the middle into a sort of hourglass form, the increasing curvature of its transverse section being balanced by a gradually increasing negative curvature in the longitudinal section. The cylinder has, in turn, been converted into an unduloid. When we hold a portion of a soft glass tube in the flame, and “draw it out,” we are in the same identical fashion converting a cylinder into an unduloid (Fig. [62]A); when on the other hand we stop the end and blow, we again convert the cylinder into an unduloid (B), but into one which is now positively, while the former was negatively curved. The two figures are essentially the same, save that the two halves of the one are reversed in the other.
Fig. 62.
That spheres, cylinders and unduloids are of the commonest occurrence among the forms of small unicellular organism, or of individual cells in the simpler aggregates, and that in the processes of growth, reproduction and development transitions are frequent from one of these forms to another, is obvious to the naturalist, and we shall deal presently with a few illustrations of these phenomena.
But before we go further in this enquiry, it will be necessary to consider, to some small extent at least, the curvatures of the six different surfaces, that is to say, to determine what modification {221} is required, in each case, of the general equation which applies to them all. We shall find that with this question is closely connected the question of the pressures exercised by, or impinging on the film, and also the very important question of the limitations which, from the nature of the case, exist to prevent the extension of certain of the figures beyond certain bounds. The whole subject is mathematical, and we shall only deal with it in the most elementary way.
We have seen that, in our general formula, the expression 1 ⁄ R + 1 ⁄ R′ = C, a constant; and that this is, in all cases, the condition of our surface being one of minimal area. In other words, it is always true for one and all of the six surfaces which we have to consider. But the constant C may have any value, positive, negative, or nil.
In the case of the plane, where R and R′ are both infinite, it is obvious that 1 ⁄ R + 1 ⁄ R′ = 0. The expression therefore vanishes, and our dynamical equation of equilibrium becomes P = p. In short, we can only have a plane film, or we shall only find a plane surface in our cell, when on either side thereof we have equal pressures or no pressure at all. A simple case is the plane partition between two equal and similar cells, as in a filament of spirogyra.
In the case of the sphere, the radii are all equal, R = R′; they are also positive, and T (1 ⁄ R + 1 ⁄ R′), or 2 T ⁄ R, is a positive quantity, involving a positive pressure P, on the other side of the equation.