V] OF FIGURES OF EQUILIBRIUM 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. 65 a). 



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 difiicult, 

 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. 



While the figures of equihbrium which are at the same time 

 surfaces of revolution are only six in number, there is an infinite 



