| a = | T | , and b² = | F | . |
| p | πp |
Hence the meridian section of the film may be traced by the focus of such a conic, if the conic is made to roll on the axis.
On the different Forms of the Meridian Line.—1. When the conic is an ellipse the meridian line is in the form of a series of waves, and the film itself has a series of alternate swellings and contractions as represented in figs. 9 and 10. This form of the film is called the unduloid.
1a. When the ellipse becomes a circle, the meridian line becomes a straight line parallel to the axis, and the film passes into the form of a cylinder of revolution.
1b. As the ellipse degenerates into the straight line joining 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 2πTm 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. Lindelöf 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.