30 M. J. Plateau on the Equilibrium-figures of Revolution 



tractcd, whilst at the same time the node becomes thinner. The 

 node now belongs neither to the catenoid nor to the unduloidj 

 but to a new ligure whose meridian curve we must examine. 



The above explanations respecting the sphere, the catenoid, 

 and the unduloid are sufficient to give an idea of the nature of 

 the reasoning applied to the equation of equilibrium ; with 

 respect to the new figure, therefore, I shall limit myself to the 

 following brief indications. 



In the first place, it must be observed that the reasoning by 

 means of which, in the case of the unduloid, the perfect sym- 

 metry of the meridian curve on opposite sides of a minimum 

 point was established, applies equally to a minimum point corre- 

 sponding to any other equilibrium-figure, and consequently to 

 that of the meridian arc of the node under consideration ; — this 

 conclusion, we may add, is confirmed by the apparent form of 

 the node. From this it follows that the complete curve has an 

 axis of symmetry perpendicular to the axis of revolution, and 

 passing through the middle of the node; so that whatever fea- 

 ture is presented by the curve on one side of the first of these 

 axes, is presented symmetrically on the other side. 



Pursuing, afterwards, the meridian curve beyond the rings, I 

 show that it passes through two points where its elements are 

 parallel to the above axis of symmetry, and I verify this deduc- 

 tion by means of a mass of oil comprised between two discs, 

 instead of two rings, the distance between the discs not exceeding 

 about the third of their diameter. By gradually subtracting oil 

 I succeed, finally, in rendering the meridian curve tangent, at 

 its two extremities, to the planes of the discs. 



I then show that the curve produced beyond these points 

 returns tow^ards the axis of symmetry, in a point of which its 

 two branches cut so as to form a loop. 



T have realized the portion of the figure generated by the 

 whole of this loop ; but before describing the experiment it will 

 be well to recall a principle, demonstrated in the second series, 

 according to which, when a surface satisfies the general condition 

 of equilibrium, it is a matter of indifference on which side of 

 this surface the liquid may be situated. In the two last experi- 

 ments just indicated, the liquid was situated on the convex side 

 of the meridian curve, so that in this case the portion of the 

 figure generated by the whole loop would form a cavity in the 

 interior of the mass of oil ; now according to the above prin- 

 ciple the oil may also be supposed to be situated on the concave 

 side of the curve, in which case the portion generated by the loop 

 will be in relief. It is thus that the figure is realized in the 

 following experiment. 



In the first place, a liquid biconvex lens is formed in a ring of 



