V] OF FIGURES OF EQUILIBRIUM 381 



oil-globule, consists of the figure containing a complete constriction, 

 then it has somewhat wide limits of stabiHty; but (2) if it contain 

 the swollen portion, then equilibrium is limited to the case of the 

 figure consisting of one complete unduloid, no less nor more; that 

 is to say when the ends of the figure are constituted by the narrowest 

 portions, and its middle by the widest portion of the entire curve. 

 The theoretical proof of this is difficult ; but if we take the proof for 

 granted, the fact itself will serve to throw light on what we have 

 learned regarding the stabihty of the cyhnder. For, when we 

 remember that the meridional section of our unduloid is generated 

 by the rolhng of an ellipse upon a straight line in its own plane, 

 we 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 outUne it approaches, and in time reaches, 

 the form of the cylinder, as a "Hmiting case"; and jpari passu, the 

 ellipse which generated it passes into a circle, as its foci come closer 

 and closer together. The cyhnder of a length equal to the circum- 

 ference of its generating circle is homologous to an unduloid whose 

 length is equal to the circumference of its generating elhpse; and 

 this is just what we recognise as constituting one complete segment 

 of the unduloid. 



The cylinder turns so easily into an unduloid, and the unduloid 

 is capable of assuming so many graded differences of form, that we 

 may expect to find it abundantly and variously represented among 

 the simpler living things. For the same reason it is the very 

 stand-by of the glass-blower, whose flasks and bottles are, of 

 necessity, unduloids*. The blown-glass bottle is a true unduloid, 

 and the potter's vase a close approach to an unduloid; but the 

 alabaster bottle, turned on the lathe, is another story. It may be 

 an imitation, or a reminiscence, of the potter's or the glass-blower's 

 work; but it is no unduloid nor any surface of minimal area at all. 



The catenoid, as we have seen, is a surface of zero pressure, and as 

 such is unlikely to form part (unless momentarily) of the closed 

 boundary of a cell. It forms a limiting case between unduloid and 

 nodoid, and, were it realised, it would seldom be visibly different from 

 the other two. In Trichodina pediculus, a minute infusorian para- 



* Unless, that is to say, their shape be cramped and their mathematical beauty 

 annihilated, by compression in a mould. 



