ing its foci,
the contracted parts of the unduloid become narrower, till at last the
figure becomes a series of spheres in contact.

In all these cases the internal pressure exceeds the external by 2T/a
where a is the semi-transverse axis of the conic. The resultant of the
internal pressure and the surface-tension is equivalent to a tension
along the axis, and the numerical value of this tension is equal to the
force due to the action of this pressure on a circle whose diameter is
equal to the conjugate axis of the ellipse.

2. When the conic is a parabola the meridian line is a catenary (fig.
11); the internal pressure is equal to the external pressure, and the
tension along the axis is equal to 2[pi]Tm where m is the distance of
the vertex from the focus.

3. When the conic is a hyperbola the meridian line is in the form of a
looped curve (fig. 12). The corresponding figure of the film is called
the nodoid. The resultant of the internal pressure and the
surface-tension is equivalent to a pressure along the axis equal to that
due to a pressure p acting on a circle whose diameter is the conjugate
axis of the hyperbola.

When the conjugate axis of the hyperbola is made smaller and smaller,
the nodoid approximates more and more to the series of spheres touching
each other along the axis. When the conjugate axis of the hyperbola
increases without limit, the loops of the nodoid are crowded on one
another, and each becomes more nearly a ring of circular section,
without, however, ever reaching this form. The only closed surface
belonging to the series is the sphere.

These figures of revolution have been studied mathematically by C.W.B.
Poisson,[3] Goldschmidt,[4] L.L. Lindelof and F.M.N. Moigno,[5] C.E.
Delaunay,[6] A.H.E. Lamarle,[7] A. Beer,[8] and V.M.A. Mannheim,[9] and
have been produced experimentally by Plateau[10] in the two different
ways already described.

[Illustration: FIG. 10.--Unduloid.]

[Illustration: FIG. 11.--Catenoid.]

[Illustration: FIG. 12.--Noboid.]

The limiting conditions of the stability of these figures have been
studied both mathematically and experimentally. We shall notice only two
of them, the cylinder and the catenoid.

[Illustration: FIG. 13.]

_Stability of the Cylinder._--The cylinder is the limiting form of the
unduloid when the rolling ellipse becomes a circle. When the ellipse
differs infinitely little from a circle, the equation of the meridian
line becomes approximatel