348 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
continuing to absorb small quantities of oil, to bring the extremities of the 
meridian line very near the edges of the solid bases fronting one another with- 
out a loss of stability in the figure, and only when they seemed to reach those 
edges was instability manifested. » 
§ 11. Since the portion of unduloid with which we are occupied has already 
reached the limit of stability when it is formed between two thin disks, and is _ 
thus free in its whole extent with the exception alone of its bases, it would be 
useless to seek to realize a portion of unduloid equally free which should extend 
on both sides beyond the centres of two constrictions, and hence we infer that 
the indefinite unduloid is, like the indefinite cylinder, an unstable figure of 
equilibrium. An experiment, hgwever, of our 2d series, affords incidentally 
an unduloid which is prolonged beyond the centres of two constrictions, but 
very close to the cylinder; to this we shall return hereafter. 
§ 12. It is now easy to see that the convex figures spoken of in § 38 of our 
2d series, while describing the formation of the liquid cylinder, figures which — 
are obtained when, after having.attached a sphere of oil to two horizontal solid 
rings equal in diameter and placed one above the other, we raise the upper 
ring by aless quantity than that which gives to the mass the cylindrical form— 
that these figures, I say, are nothing else but portions of the dilatations of the 
unduloid; only, when these convex figures are produced by the process just 
recalled, they are so placed that their axis is vertical.* 
Let us conceive, in effect, an unduloid realized by means of two thick disks, 
(§ 10,) and consequently in a state of stable equilibrium, and imagine that we 
place at equal distances to the right and left of the middle of this figure, be- 
tween that middle and the thick disks, two vertical solid rings, having their 
centres on the axis and their exterior circumference precisely at the surface of 
the mass; it is clear that these rings will not destroy the equilibrium of the 
figure. Now, if we suppose that the parts of the figure situated beyond these 
rings are replaced by convex spherical caps resting on the latter, and whose 
curvature is such that it occasions a pressure equal to that which pertains to 
the rest of the figure, equilibrium will still evidently exist, and it will still be 
perfectly stable, since the distance of the rings is less than that which corre- 
sponds to the limit of stability. But, then, if the rings are not sufficiently 
separated for the portion of the meridian line which extends from one to the 
other to contain points of inflection, it is evident that the whole will constitute 
one of the convex figures in question; for, according to the different forms of 
the unduloid, the meridian line of the portion comprised between the rings may 
vary from an are of a circle, with its centre on the axis, to a straight line, as 
in these convex figures. For these last not to be portions of an unduloid, it 
would be necessary that between the same rings, placed at the same distance 
from one another, and with an equal mass of oil, there should be two figures 
of equilibrium possible, both of them stable, which experiment contradicts. If, 
after having transformed a sphere of oil into one of the convex figures in ques- 
tion, whether by increasing the separation of the rings or by subtracting a cer- 
tain quantity of the liquid, we agitate the alcoholic mixture so as to give con- 
siderable motion to the mass of oil, but still not enough to disunite it, and then — 
allow it to return to a state of rest, it will always resume identically the same 
form. 
In the experiments of §§ 44 and 45 of the 2d series, tvhen the rings or disks 
were placed at a distance of four times their diameter, and the liquid mass com- 
prised between them was sufficient for the stability of the figure, this figure 
evidently constituted part of an unduloid; but as, by the abstraction of oil, we 
afterwards arrived, through a very small diminution of the mass, at its sponta- 
neous destruction, it follows that the portion of unduloid in question was but 
we en Bien Rte oR ie rs hd neal I oe 
* One of these convex figures is represented at Fig. 21 of 2d series. 
