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



we may conclude that the second limit of the variations of a 

 complete iinduloid consists of an indefinite series of equal spheres 

 touching each other on the axis. 



In each of these experiments is realized that portion of a com- 

 ])lcte unduloid which is comprised between the middle points 

 of two successive nodes, so that the figure of oil is composed of 

 a whole ventral segment {renflement) adjacent to two semi-nodes 

 (demi-etranrjlemcnts). I have also realized a portion of an undu- 

 loid consisting of an entire node between two portions of ventral 

 scments. For this purpose I attach a sufficiently large mass of 

 oil to two solid vertical rings, 7 centims. in diametei-, placed 

 opposite to each other at a distance of 11 centims. asunder. By 

 o-radually absorbing portions of liquid, the figure soon becomes 

 cylindrical with convex bases, and afterwards narrower and nar- 

 rower in the middle. By operating cautiously, I can reduce the 

 diameter of a node, at its middle point, to 27 millims. without 

 causing the figure to lose its stability. As to the bases of the 

 figure, they remain convex. I remark that in this experiment 

 we witness the variations of the unduloid from the cylinder itself, 

 which, as above mentioned, forms one of its limits. 



From the fact that portions of an unduloid more extended 

 than those above described cannot be realized, I conclude that 

 the unduloid, like the cylinder, has a limit of stability. 



1 resume the experiments with the solid cylinder. If, vvhilst 

 the diameter of this cylinder remains the same, the mass of oil 

 be gradually increased, the liquid figure will at the same time 

 increase more and more in length and in equatorial thickness. 

 In this kind of variation the minima distances from the axis do 

 not change, for they are always equal to the radius of the solid 

 cylinder ; but at the point of the meridian curve which is the 

 most distant from the axis, that is to say at the equator of the 

 realized figure, the radius of curvature and the normal both 

 increase. Let us now suppose that whenever a fresh quantity of 

 oil is added to the mass, one (say the left) extremity of the same 

 is always brought to the same position on the cylinder, so that the 

 right extremity alone recedes. If we imagine these variations 

 pushed to their limit, it is clear that all the nodes of the complete 

 meridian curve will then have receded to infinity, or, in other 

 words, have disappeared, with the exception of that which corre- 

 sponds to the left extremity of the figure of oil, and that this 

 meridian curve will then be reduced to one having two infinite 

 branches — like the parabola — having its vertex turned towards 

 the axis of revolution, and its axis of symmetry perpendicular 

 to the same. 



It is, however, easy to determine the nature of this curve; for 

 since, as above remarked, the radius of curvature and the normal 



