brium only when the portion
considered is such that the tangents to the catenary at its extremities
intersect before they reach the directrix.

To prove this, let us consider the catenary as the form of equilibrium
of a chain suspended between two fixed points A and B. Suppose the chain
hanging between A and B to be of very great length, then the tension at
A or B will be very great. Let the chain be hauled in over a peg at A.
At first the tension will diminish, but if the process be continued the
tension will reach a minimum value and will afterwards increase to
infinity as the chain between A and B approaches to the form of a
straight line. Hence for every tension greater than the minimum tension
there are two catenaries passing through A and B. Since the tension is
measured by the height above the directrix these two catenaries have the
same directrix. Every catenary lying between them has its directrix
higher, and every catenary lying beyond them has its directrix lower
than that of the two catenaries.

Now let us consider the surfaces of revolution formed by this system of
catenaries revolving about the directrix of the two catenaries of equal
tension. We know that the radius of curvature of a surface of revolution
in the plane normal to the meridian plane is the portion of the normal
intercepted by the axis of revolution.

The radius of curvature of a catenary is equal and opposite to the
portion of the normal intercepted by the directrix of the catenary.
Hence a catenoid whose directrix coincides with the axis of revolution
has at every point its principal radii of curvature equal and opposite,
so that the mean curvature of the surface is zero.

The catenaries which lie between the two whose direction coincides with
the axis of revolution generate surfaces whose radius of curvature
convex towards the axis in the meridian plane is less than the radius of
concave curvature. The mean curvature of these surfaces is therefore
convex towards the axis. The catenaries which lie beyond the two
generate surfaces whose radius of curvature convex towards the axis in
the meridian plane is greater than the radius of concave curvature. The
mean curvature of these surfaces is, therefore, concave towards the
axis.

Now if the pressure is equal on both sides of a liquid film, and if its
mean curvature is zero, it will be in equilibrium. This is the case with
the two catenoids. If the mean curvature is convex towards the axis