covers certain cases included under, a much more important and fundamental rule, due not to Sachs but to Errera; that (3) the incipient partition-wall of a dividing cell tends to be such that its area is the least possible by which the given space-content can be enclosed.
Let us return to the case of our cube, and let us suppose that, instead of bisecting it, we desire to shut off some small portion only of its volume. It is found in the course of experiments upon soap-films, that if we try to bring a partition-film too near to one side of a cubical (or rectangular) space, it becomes unstable; and is easily shifted to a totally new position, in which it constitutes a curved cylindrical wall, cutting off one corner of the cube. It meets the sides of the cube at right angles (for reasons which we have already considered); and, as we may see from the symmetry {349} of the case, it constitutes precisely one-quarter of a cylinder. Our plane transverse partition, wherever it was placed, had always the same area, viz. aÿ£¢2ã₤; and it is obvious that a cylindrical wall, if it cut off a small corner, may be much less than this. We want, accordingly, to determine what is the particular volume which might be partitioned off with equal economy of wall-space in one way as the other, that is to say, what area of cylindrical wall would be neither more nor less than the area aÿ£¢2ã₤. The calculation is very easy.
The surface-area of a cylinder of length a is 2ürã₤ôñã₤a, and that of our quarter-cylinder is, therefore, aã₤ôñã₤ürã₤ãã₤2; and this being, by hypothesis, =ã₤aÿ£¢2ã₤, we have a =ã₤ürã₤ãã₤2, or r =ã₤2aã₤ãã₤ü.
The volume of a cylinder, of length a