soon the ends became gradually concave, and the concavity deepened as more and more oil was withdrawn, until at a certain limit, the whole oil-globule broke up in general disruption. We learn from this that the limiting size of the catenoid was reached when the distance of the supporting rings was to their diameter as 47 to 71, or, as nearly as possible, as two to three; {229} and as a matter of fact it can be shewn that 2/3 is the true theoretical value. Above this limit of 2/3, the inevitable convexity of the end-surfaces shows that a positive pressure inwards is being exerted by the surface film, and this teaches us that the sides of the figure actually constitute not a catenoid but an unduloid, whose spontaneous changes tend to a form of greater stability. Below the 2/3 limit the catenoid surface is essentially unstable, and the form into which it passes under certain conditions of disturbance such as that of the excessive withdrawal of oil, is that of a nodoid (Fig. 65A). The unduloid has certain peculiar properties as regards its limitations of stability. But as to these we need mention two facts only: (1) that when the unduloid, which we produce with our soap-bubble or our oil-globule, consists of the figure containing a complete constriction, it has somewhat wide limits of stability; but (2) if it contain the swollen portion, then equilibrium is limited to the condition that the figure consists simply of one complete unduloid, that is to say that its ends are constituted by the narrowest portions, and its middle by the widest portion of the entire curve. The theoretical proof of this latter fact is difficult, but if we take the proof for granted, the fact will serve to throw light on what we have learned regarding the stability of the cylinder. For, when we remember that the meridional section of our unduloid is generated by the rolling of an ellipse upon a straight line in its own plane, we shall easily see that the length of the entire unduloid is equal to the circumference of the generating ellipse. As the unduloid becomes less and less sinuous in outline, it gradually approaches, and in time reaches, the form of a cylinder; and correspondingly, the ellipse which generated it has its foci more and more approximated until it passes into a circle. The cylinder of a length equal to the circumference of its generating circle is therefore precisely homologous to an unduloid whose length is equal to the circumference of its generating ellipse; and this is just what we recognise as constituting one complete segment of the unduloid. ――――――――――