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. a2 ; 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 a2 . The calculation is very easy.
The surface-area of a cylinder of length a is 2πr · a, and that of our quarter-cylinder is, therefore, a · πr ⁄ 2; and this being, by hypothesis, = a2 , we have a = πr ⁄ 2, or r = 2a ⁄ π.
The volume of a cylinder, of length a, is aπr2 , and that of our quarter-cylinder is a · πr2 ⁄ 4, which (by substituting the value of r) is equal to a3 ⁄ π.
Now precisely this same volume is, obviously, shut off by a transverse partition of area a2 , if the third side of the rectangular space be equal to a ⁄ π. And this fraction, if we take a = 1, is equal to 0·318..., or rather less than one-third. And, as we have just seen, the radius, or side, of the corresponding quarter-cylinder will be twice that fraction, or equal to ·636 times the side of the cubical cell.
Fig. 137.
If then, in the process of division of a cubical cell, it so divide that the two portions be not equal in volume but that one portion by anything less than about three-tenths of the whole, or three-sevenths of the other portion, there will be a tendency for the cell to divide, not by means of a plane transverse partition, but by means of a curved, cylindrical wall cutting off one corner of the original cell; and the part so cut off will be one-quarter of a cylinder.
By a similar calculation we can shew that a spherical wall, cutting off one solid angle of the cube, and constituting an octant of a sphere, would likewise be of less area than a plane partition as soon as the volume to be enclosed was not greater than about {350} one-quarter of the original cell[387]. But while both the cylindrical wall and the spherical wall would be of less area than the plane transverse partition after that limit (of one-quarter volume) was passed, the cylindrical would still be the better of the two up to a further limit. It is only when the volume to be partitioned off {351} is no greater than about 0·15, or somewhere about one-seventh, of the whole, that the spherical cell-wall in an angle of the cubical cell, that is to say the octant of a sphere, is definitely of less area than the quarter-cylinder. In the accompanying diagram (Fig. [138]) the relative areas of the three partitions are shewn for all fractions, less than one-half, of the divided cell.
Fig. 138.