366 Prof. J. Plateau on the Figures of Equilibrium 



at its limit of stability, has a length equal to the circumference 

 of the rolling circle. But this circumference is evidently equal 

 to that of the cylinder ; hence the limiting cylinder has a length 

 equal to its own circumference ; and hence, lastly, the ratio of 

 the length to the diameter has the precise value it. 



I have already discussed the question of the limit of stability 

 of the catenoid in my Tenth Series, and I then described expe- 

 riments which fully confirm the theoretical result. 



No general statement can be given of the limit of stability of 

 the nodoid (nodo'ide), whether the figure be generated by a por- 

 tion of a node of the meridian line, or by an arc of this line 

 convex towards the outside. 



Except in the case of the cylinder, the partial figure of revolu- 

 tion can be contained between unequal circular bases ; and then 

 the conditions of stability are necessarily different. We have a 

 curious instance of this in the case of the catenoid : if the cir- 

 cular section of the neck be taken as one of the bases, the figure 

 has no longer any limit of stability ; that is to say, the other 

 base may be taken as far away as we please in the indefinite 

 figure without the portion contained between the two bases 

 ceasing to be stable. I verify this conclusion experimentally 

 upon a catenoid of which the circle at the neck had a diameter 

 of only 3*5 centims., while the diameter of the other base was 

 20 centims. 



I next consider from a general point of view the question of 

 the stability of figures of equilibrium. It is admitted by geo- 

 metricians, as a result of analysis, that the surfaces represented 



by the equation ^ -f — =C (that is to say, surfaces of constant 



mean curvature), are also those which, for a given volume, have 

 the smallest superficial area. But if we were to accept this prin- 

 ciple without restriction, it would follow that every partial liquid 

 figure of equilibrium terminated by a solid system would be ne- 

 cessarily stable, whatever portion it might represent of the com- 

 plete figure : the cylinder, for instance, would be completely 

 stable however great its length ; the unduloid likewise would 

 remain perfectly stable whatever the number of bulging and 

 constricted portions contained between its two solid bases, &c. In 

 fact the superficial layer, being, as is now known, really in a 

 state of tension, makes a constant eifort to contract; hence, if its 

 area were always a minimum in a state of equilibrium, any very 

 small change of form would increase its area, and consequently 

 the superficial layer would tend to resume its former dimensions 

 and to restore the figure of equilibrium. 



Geometricians have been led to the above principle by the ana- 

 lytical result that the variation of such surfaces is always zero, 



