368 FIGURES OF EQUILIBRIUM OF A LIQUID MASS 
one single node: it is formed, as will be seen by Fig. 40, of the are which unites 
two consecutive nodes and of two portions of the latter. 
§ 37. The variations of the nodoid, finally, have, like those of the unduloid, 
a third limit, which is disclosed by the same experiments that have led us to a 
knowledge of the nodoid itself. In the experiments of §§ 20 and 22, when, after 
having formed a cylinder between two rings placed at a less distance than % of 
their diameter, we progressively remove some of the liquid, the partial figure, 
as we have seen, becomes first an unduloid, then by degrees attains the catenoid, 
after which it immediately passes into the nodoid; whence it evidently follows 
that the catenoid is one of the limits of the variations of the nodoid, and, more- 
over, that it constitutes a new transition from the latter to the unduloid. We 
have already recognized (§ 34) another, consisting in an indefinite succession of 
spheres. 
Phe third limit, then, of the variations of the nodoid is the catenoid, and it is 
easily made apparent how the figure becomes thus modified. If we consider 
that the experiments just spoken of realize the portion of the nodoid generated 
by an are pertaining to a node, and presenting its concavity externally, we shall 
thence conclude that the portion of the nodoid which passes into the catenoid 
is that which is generated by one of the nodes, whose summit becomes that of 
the meridian catena. 'This being premised, let us conceive that each of the nodes 
of the complete meridian line becomes gradually modified to arrive at the catena, 
and let us imagine, for the sake of distinctnéss, that, during all these modifica- 
tions, the distance of the summits from the axis of revolution remains constant. 
, aot HORT 
In proportionas the nodes approach the catena the quantity Mo N tends towards 
zero, but on all the ares which unite the nodes with one another the quantities 
I es ; 
M and N are of the same sign, and consequently the quantity Mon” relation to 
these ares cannot tend towards zero unless M and N tend at the same time 
towards infinity; all the points, then, of these ares will withdraw indefinitely 
from the axis of revolution, while their curvature becomes at the same time 
indefinitely weaker; whence it follows that the extremities of the nodes will 
withdraw further and further from the axis, while, by the increasing develop- 
ment of the intermediate ares, which, from the nature of the curve, evidently 
cannot diminish in curvature without acquiring greater extension, the nodes 
will separate more and more from one another, until, at the limit, they are all 
infinitely distant and infinitely elongated. If, then, we consider one in partic- 
ular, the whole curve will be reduced to that one alone, and, on the other hand, 
its extremity or point will have disappeared, and it will be found to be trans- 
formed into the meridian line of a catenoid, that is to say into a catena. 
§ 38. A last question now presents itself: Are there other figures of equili- 
brium of revolution besides those of which we have thus far recognized the ex- 
istence? All these last are such that portions of them can always be comprised 
between two equal and parallel disks. Now our experiments have exhausted 
all the combinations of that kind; whence we must conclude that if there were 
still other figures, they would be of such a nature as not to be capable of fulfilling 
that condition, and, for that, it would evidently be necessary that their meridian 
lines should present no point whose distance from the axis of revolution would 
be a maximum ora minimum. As these lines, moreover, could not reach the 
axis, (§ 2,) they must continue always to leave it, from a first point situated at 
infinity on an asymptote parallel to that axis, up to another point situated like- 
wise at infinity. This being so, at the first of these two extreme points, the 
radius of curvature would be necessarily infinite, while the perpendicular would 
BH 1 
be finite, and the equation of equilibrium would be reduced tox=C; but from 
