Page:Encyclopædia Britannica, Ninth Edition, v. 5.djvu/79

Rh CAPILLARY ACTION 67 principal radii of curvature, though here proved only in the case of a surface of revolution, must be true of all surfaces. For the curvature of any surface at a given point may be completely denned in terms of the positions of its principal normal sections and their radii of curvature. Before going further we may deduce from equation 9 the nature of all the figures of revolution which a liquid film can assume. Let us first determine the nature of a curve, such that if it is rolled on the axis its origin will trace out the meridian section of the bubble. Since at any instant the rolling curve is rotating about the point of contact with the axis, the line drawn from this point of contact to the tracing point must be normal to the direction of motion of the tracing point. Hence if N is the point of contact, NP must be normal to the traced curve. Also, since the axis is a tangent to the rolling curve, the ordinate PR is the perpendicular from the tracing point P on the tangent. Hence the relation between the radius vector and the perpendicular on the tangent of the rolling curve must be identical with the relation between the normal PN and the ordinate PR of the traced curve. If we write r for PN, then y = r cos. a, and equation 9 becomes pr This relation between y and r is identical with the relation between the perpendicular from the focus of a conic section on the tangent at a given point and the focal distance of that point, provided the transverse and con jugate axes of the conic are 2a and 2b respectively, where a =, and 2 =. P *P 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 LlNE. (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. 8 and 9. This form of the film is called the un- duloid. (la.) 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. (16.) As the ellipse degenerates into the straight line joining its foci, the contracted pp.rts 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 2T external by where a is the semitransverse axis of the ct 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. 10), the internal pressure is equal to the external pressure, and the tension along the axis is equal to 27rTm 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. 11). 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 mathe matically by Poisson, 1 Goldschmidt, 2 Lindelof and Moigno, 3 Delaunay, 4 Lamarle, 5 Beer, 6 and Mannheim, 7 and have been produced experimentally by Plateau 8 in the two different ways already described. Fia. 9. Unduloid. FIG. 10. Catenoid. FIG. 11. Nodoid. 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. 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 3/ meridian line becomes approximately y = a + c sin. - d where c is small. This is a simple harmonic wave-line, whose mean distance from the axis is a, whose wave-length is 2ira, and whose amplitude is c. The internal pressure T corresponding to this unduloid is as before p=. Now consider a portion of a cylindric film of length x terminated by two equal disks of radius r and con taining a certain volume of air. Let one of these disks be made to approach the other by a small pj g 12. quantity dx. The film will swell out into the convex part of an unduloid, having its largest section midway between the disks, and we have to determine whether the internal pressure will be greater or less than before. If A and C (fig. 12) are the disks, and if x the distance between the disks is equal to irr half the wave-length of the harmonic curve, the disks will be at the points where the curve is at its mean distance from the T axis, and the pressure will therefore be as before. If Aj, GJ are. the disks, so that the distance between them is less than irr, the curve must be produced beyond the disks before it is at its mean distance from the axis. Hence in this case the mean distance is less than r, and T the pressure will be greater than. If, on the other hand, the disks are at A 2 and C 2, so that the distance between them is greater than irr, the curve will reach its mean dis- 1 Nouvelle theorie de V action capillaire (1 831). 2 Determinatio superficiei minima; rotatione curvce data d uo puncta jungentis circa datum axein ortce (Gottingen, 1831). 3 Lecons de calcul des variations (Paris, 1861). 4 &quot; Sur la surface de revolution dont la courbure moyenne est constante,&quot; Lioumlles Journal, vi. 5 &quot; Theorie georaetrique des rayons el centres de courbure,&quot; Bullet, de FA cad. de Belgique, 1857. 6 Tractatus de Theoria MaUiematica Pkcenomenorum in LiquiJis actioni gravitatis detractis observatorum (Bonn, 1857). 7 Journal I lnstitut, No. 1260. 8 Statique experimentale et Iheorique des Uguides. A A, C, C c, % B ~ J ~-J