WITHDRAWN FROM THE ACTION OF GRAVITY. 361 
fraction of the distance in question and nearly three-fourths of that distance. 
The complete figure with which we are occupied is thus not always similar to 
itself, as are the sphere, the cylinder, or the catenoid; like the unduloid it is 
susceptible of variations of form. A comparison of the liquid figures repre- 
sented by Figs. 25 and 26 leads to the same conclusion. 
§ 29. Before proceeding, we will notice a remarkable particular. If we sup- 
pose the node in relief, the liquid which occupies it is in the coneavity of the 
curve; and since this line does not change the direction of its curvature in pass- 
ing the point u, (Fig. 30,) the liquid will still occupy the concavity of each 
of the prolongations wv and ww; it fills therefore the spaces comprised between 
these prolongations and the node, so that this node is engaged, whether com- 
pletely or partially, in the interior of the mass. If we suppose the node hollow, 
(en creux,) it is, as may be easily seen, the prolongations wv and ww which are 
then engaged in the liquid. Hence results this singular consequence, that, 
though the general condition of equilibrium is satisfied, we cannot represent to 
ourselves the complete figure, except in the state of a simple surface, not in that 
of a liquid mass. In this last state it is only possible to imagine isolated por- 
tions of the figure—such, for instance, as the portion generated by the node alone. 
This peculiarity of a surface re-entering into the mass is one of those to which 
allusion was made in §1 of the second series, and which would render the 
realization of certain figures of equilibrium in their whole extent impossible, 
even if those figures did not extend to infinity. 
§ 30. Let us attempt now to discover the course of the curve beyond the 
points » and w, (Fig. 30.) We already know, from reasons stated in § 26, and 
illustrated by Fig. 28, that as long as the branches of the curve continue to 
retire from the axis of revolution, the curvature cannot change its direction, and 
consequently remains concave towards that axis. This being so, there are evi- 
dently but three hypotheses possible: either the branches in question retire from 
the axis of revolution in such manner that their distance from the latter tends 
towards infinity, or they tend towards an asymptote parallel to this axis; or 
each of them presents, at a finite distance from the point w of the node, (Fig. 30,) 
a point at which the element is parallel to the same axis. The first of these _ 
we may at once dismiss ; it would require, as has been already shown, (§ 24,) 
that at the points situated at infinity on the two branches, the radius of curva- 
ture and the perpendicular should be both infinite, and thus the quantity 
1 
MIN would be equal to zero. 
Let us examine, then, the second hypothesis, that, namely, of an asymptote paral- 
lel to the axis of revolution. At the point x (Fig. 30) the perpendicular is in- 
finite, and the radius of curvature finite, (§26;) at the point where the branch 
nuv prolonged would reach the asymptote, on the contrary, the radius of curva- 
ture would be infinite, and the perpendicular, which would measure the distance 
from that point to the axis, would be finite. In passing, then, from the point 
to this extreme point, the perpendicular, at first superior in length to the radius 
of curvature, would afterwards become inferior to it; whence it follows that 
there would be on the curve a point where the perpendicular and the radius of cur- 
vature would be equal, and for which consequently the centre of curvature would 
be on the axis of revolution. Let a be this point, o the corresponding centre of 
curvature, and af a small are of a circle described from the point o as a centre. 
One branch of the curve would quit the point a in the same direction and with 
the same curvature as the are af, and would then immediately separate from the 
latter. Now let us suppose that at its departure from a, the curvature should 
at first go on decreasing; the curve will, at commencing, be necessarily exterior 
to the are of a circle. Let ay be a small are of this curve, in the whole extent 
of which the curvature decreases, and“let the length of the are af be taken 
