32 j\I. J. Plateau on the Equilibrium-figures of Revolution 



consequence of this form I give to tbe generated figure the name 

 nodoid (nodo'ide). 



The nodoid ])rcscnts a remarkable peculiarity : it cannot, 

 in its complete state, be conceived to be formed by a liquid; in 

 fact, from tbe form of tbe meridian curve, it is at once seen tbat 

 the portions generated by tbe loops would be imbedded in the 

 interior of tbe mass ; but these same portions, as well as other 

 suitably chosen ones, may, as I have described, be isolated. 



On performing tbe above experiments, by means of which 

 these portions are realized, it becomes evident that the nodoid, 

 like the unduloid, is susceptible of variations. Tbe following 

 example will elucidate my meaning. In the experiment which 

 commences with a biconvex lens, the thickness of the same may, 

 before piercing, be considerably reduced ; tbe aperture thus 

 made then widens spontaneously, the meridian loop becomes 

 much shorter, and its point much more blunt. Thus, the 

 points of the loops of tbe complete meridian curve remaining at 

 the same distance from the axis of revolution, the loops them- 

 selves may assume very diflPerent forms. 



On examining these variations and their limits, I arrive at the 

 following results : — 



In a first mode of variation, if we suppose, for simplicity, that 

 the distance between the axes of symmetry of tbe loops remains 

 constant, these loops become smaller and smaller, their vertices 

 approach the axis of revolution, and at the same time the curva- 

 ture of the intermediate arcs approaches more and more the 

 circular phase, ultimately the loops vanish, the points which form 

 their vertices attain the axis of revolution, and the arcs which 

 connected them become semicircles, so tbat the generated figure 

 is an indefinite series of equal spheres touching each other on the 

 axis. We have seen that this series of spheres is also one of 

 the limits of the variations of the unduloid ; it forms therefore 

 the phase of transition from the unduloid to the nodoid. 



In a second mode of variation, if, for simplicity, we conceive 

 the length of the loops — that is to say tbe distance from tbe 

 vertex to the point of each — to remain constant, the curve 

 recedes more and more from the axis of revolution, or rather, 

 since attention is directed to the latter, the axis recedes from 

 the curve ; at the same time the loops become broader, approach, 

 and afterwards project over each other, until finally, when the 

 axis of revolution is at an infinite distance, and therefore ceases 

 to exist, the whole meridian curve becomes condensed into the 

 circumference of one circle, and the generated figure is a cylinder 

 placed transversely with respect to the axes of revolution of the 

 nodoids of which it is the limit. 



In a last mode of variation, if — again for simplification — we 



