which are neither Spherical nor CylimticaL 29 



corresponding to the equator of the ventral segment increase as 

 the volume of the mass of oil is augmented, it is clear that they 

 and this volume will become infinite at the same time ; at this 



limit, therefore, the quantity tTj- + j;jj which is constant at all 



points of the meridian curve, is equal to zero. But from this 

 we deduce the relation M=— N, which shows that at each 

 point of the curve tlie radius of curvature is equal and opposite 

 to the normal. Geometricians are well aware that the catenary 

 is the only curve which enjoys this property. 



From this it follows that the variations of the unduloid have 

 a third limit, which is a new equilibrium-figure whose meridian 

 curve is a catenary with its vertex turned towards the axis of 

 revolution, and its radius of curvature at this vertex equal and 

 opposite to the distance between the same and the axis. The 

 form of its meridian curve here suggests the name catenoid for 

 the corresponding equilibrium -figure. 



In the last experiment above recorded, it was only by an 

 imaginary extension of the operation that we arrived at the cate- 

 noid ; a portion of this new figure, however, may be easily real- 

 ized in the following manner: — In the first place a cyhnder is 

 formed between two solid rings, whose distance asunder does not 

 exceed two-thirds of their common diameter, afterwards liquid is 

 gradually withdrawn from the mass. In this manner, whilst the 

 portion between the two rings becomes hollowed out more deeply 

 to form a node, the bases are seen to lose gradually their con- 

 vexity, until finally they become altogether plane. Now, when 

 the figure of the oil has arrived at this phase, we have on the 



bases w + ^ = ; for considering the plane as a surface of revo- 



]\I N or 



lution, the radius of curvature and the normal are evidently in- 

 finite at each of its points ; but equilibrium requires that the 



quantity vr + ^ should have the same value at all points of the 



^ •' M N 2 ]^ 



realized surface, consequently on the node we have ^ + aj~^' 



which relation, as we have just seen, characterizes the catenoid. 

 In the memoir, I show that as long as the bases have not 

 entirely lost their curvature the node belongs to the unduloid ; 

 in this' experiment, therefore, we witness the transition of a por- 

 tion of an unduloid from the cylinder to the catenoid. 



When the distance between the rings is much less than two- 

 thirds of their diameter, when, for example, it is reduced to the 

 half or the third of this diameter, then, after arriving at the 

 plane bases, if we continue to absorb the oil, these bases arc 

 observed to become more and more concave as liquid is sub- 



