Let us imagine a liquid drop which by appropriate conditions has been made to assume the form of a cylinder; we have already seen that its ends will be terminated by portions of spheres. Since one and the same liquid film covers the sides and ends of the drop (or since one and the same delicate membrane encloses the sides and ends of the cell), we assume the surface-tension (_T_) to be everywhere identical; and it follows, since the internal fluid-pressure is also everywhere identical, that the expression (1/_R_ + 1/_R′_) for the cylinder is equal to the corresponding expression, which we may call (1/_r_ + 1/_r′_), in the case of the terminal spheres. But in the cylinder 1/_R′_ = 0, and in the sphere 1/_r_ = 1/_r′_. Therefore our relation of equality becomes 1/_R_ = 2/_r_, or _r_ = 2 _R_; that is to say, the sphere in question has just twice the radius of the cylinder of which it forms a cap. [Illustration: Fig. 67.] And if _Ob_, the radius of the sphere, be equal to twice the radius (_Oa_) of the cylinder, it follows that the angle _aOb_ is an angle of 60°, and _bOc_ is also an angle of 60°; that is to say, the arc _bc_ is equal to (1/3) π. In other words, the spherical disc which (under the given conditions) caps our cylinder, is not a portion taken at haphazard, but is neither more nor less than that portion of a sphere which is subtended by a cone of 60°. Moreover, it is plain that the height of the spherical cap, _de_, = _Ob_ − _ab_ = _R_ (2 − √3) = 0·27 _R_, where _R_ is the radius of our cylinder, or one-half the radius of our spherical cap: in other words the normal height of the spherical cap over the end of the cylindrical cell is just a very little more than one-eighth of the diameter of the cylinder, or of the radius of the {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 equilibrium 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 [Illustration: Fig. 68.]