{227} sphere. And these are the proportions which we recognise, under normal circumstances, in such a case as the cylindrical cell of Spirogyra where its free end is capped by a portion of a sphere.
Among the many important theoretical discoveries which we owe to Plateau, one to which we have just referred is of peculiar importance: namely that, with the exception of the sphere and the plane, the surfaces with which we have been dealing are only in complete equiôÙlibôÙrium within certain dimensional limits, or in other words, have a certain definite limit of stability; only the plane and the sphere, or any portions of a sphere, are perfectly stable, because they are perfectly symmetrical, figures. For experimental demonstration, the case of the cylinder is the simplest. If we produce a liquid film having the form of a cylinder, either by
drawing out a bubble or by supporting between two rings a globule of oil, the experiment proceeds easily until the length of the cylinder becomes just about three times as great as its diameter. But somewhere about this limit the cylinder alters its form; it begins to narrow at the waist, so passing into an unduloid, and the deformation progresses quickly until at last our cylinder breaks in two, and its two halves assume a spherical form. It is found, by theoretical conôÙsiôÙdeôÙraôÙtions, that the precise limit of stability is at the point when the length of the cylinder is exactly equal to its circumference, that is to say, when L =ã₤2üR,