the
film will move from the axis. Hence if a film in the form of the
catenoid which is nearest the axis is ever so slightly displaced from
the axis it will move farther from the axis till it reaches the other
catenoid.

If the mean curvature is concave towards the axis the film will tend to
approach the axis. Hence if a film in the form of the catenoid which is
nearest the axis be displaced towards the axis, it will tend to move
farther towards the axis and will collapse. Hence the film in the form
of the catenoid which is nearest the axis is in unstable equilibrium
under the condition that it is exposed to equal pressures within and
without. If, however, the circular ends of the catenoid are closed with
solid disks, so that the volume of air contained between these disks and
the film is determinate, the film will be in stable equilibrium however
large a portion of the catenary it may consist of.

The criterion as to whether any given catenoid is stable or not may be
obtained as follows:--

[Illustration: FIG. 14.]

Let PABQ and ApqB (fig. 14) be two catenaries having the same directrix
and intersecting in A and B. Draw Pp and Qq touching both catenaries, Pp
and Qq will intersect at T, a point in the directrix; for since any
catenary with its directrix is a similar figure to any other catenary
with its directrix, if the directrix of the one coincides with that of
the other the centre of similitude must lie on the common directrix.
Also, since the curves at P and p are equally inclined to the directrix,
P and p are corresponding points and the line Pp must pass through the
centre of similitude. Similarly Qq must pass through the centre of
similitude. Hence T, the point of intersection of Pp and Qq, must be the
centre of similitude and must be on the common directrix. Hence the
tangents at A and B to the upper catenary must intersect above the
directrix, and the tangents at A and B to the lower catenary must
intersect below the directrix. The condition of stability of a catenoid
is therefore that the tangents at the extremities of its generating
catenary must intersect before they reach the directrix.

_Stability of a Plane Surface._--We shall next consider the limiting
conditions of stability of the horizontal surface which separates a
heavier fluid above from a lighter fluid below. Thus, in an experiment
of F. Duprez ("Sur un cas particulier de l'equilibre des liquides,"
_Nouveaux Mem. del' Acad. de Belgique, 1851