58 CAPILLARY ACTION The condition of equilibrium is that this expression (which we may for the sake of distinctness call the potential energy) shall be a minimum. This condition when worked out gives not only the equation of the free surface in the form already established by Laplace, but the conditions of the angle of contact of this surface with the surface of a solid. Gauss thus supplied the principal defect in the great work of Laplace. He also pointed out more distinctly the nature of the assumptions which we must make with respect to the law of action of the particles in order to be consistent with observed phenomena. He did not, however, enter into the explanation of particular phenomena, as this had been done already by Laplace. He points out, however, to physicists the advantages of the method of Segner and Gay-Lussac, afterwards carried out by Quincke, of measur ing the dimensions of large drops of mercury on a horizontal or slightly concave surface, and those of large bubbles of air in transparent liquids resting against the under side of a horizontal plate of a substance wetted by the liquid. In 1831 Poisson published his Nouvelle Theorie de I Action Capillaire. He maintains that there is a rapid variation of density near the surface of a liquid, and he gives very strong reasons, which have been only strength ened by subsequent discoveries, for believing that this is the case. He then proceeds to an investigation of the equilibrium of a fluid on the hypothesis of uniform density, and he arrives at the conclusion that on this hypothesis none of the observed capillary phenomena wo aid take place, and that, therefore, Laplace s theory, in which the density is supposed uniform, is not only insufficient but erroneous. In particular he maintains that the constant pressure K, which occurs in Laplace s theory, and which on that theory is very large, must be in point of fact very small, but the equation of equilibrium from which he concludes this is itself defective. Laplace assumes that the liquid has uniform density, and that the attraction of its molecules extends to a finite though insensible distance. On these assumptions his results are certainly right, and are confirmed by the independent method of Gauss, so that the objections raised against them by Poisson fall to the ground. But whether the assumption of uniform density be physically correct is a very different question, and Poisson has done good service to science in showing how to carry on the investigation on the hypothesis that the density very near the surface is different from that in the interior of the fluid. The result, however, of Poisson s investigation is practi cally equivalent to that already obtained by Laplace. In both theories the equation of the liquid surface is the same, involving a constant H, which can be determined only by experiment. The only difference is in the manner in which this quantity H depends on the law of the molecular forces and the law of density near the surface of the fluid, and as these laws are unknown to us we cannot obtain any test to discriminate between the two theories. We have HOW described the principal forms of the theory of capillary action during its earlier development. In more recent times the method of Gauss has been modified so as to take account of the variation of density near the surface, and its language has been translated in terms of the modern doctrine of the conservation of energy. 1 M. Plateau, 2 who has himself made the most elaborate study of the phenomena of surface-tension, has adopted the 1 See Prof. Betti, Teoria della Capillaritd : Nuovo Cimento, 1867 ; a memoir by M. Stahl, " Ueber einige Punckte in der Theorie der Capillarerscheinungen," Pogg. Ann., cxxxix. p. 239 (1870) ; and M. Van der Waal s Over de Continviteit van den Gas- en Vloeistoftoe- stwid. The student will find a good account of the subject from a mathematical point of view in Professor Challis s " Report on the Theory of Capillary Attraction, Brit. Ass. Report, iv. p. 253 (1834.) ^ M, Plateau, Statiq>M experimentale et Iheorique des liquides. following method of getting rid of the effects of gravity : He forms a mixture of alcohol and water of the same density as olive oil. He then introduces a quantity of oil into the mixture. It assumes the form of a sphere under the action of surface-tension alone. He then, by means of rings of iron-wire, disks, and other contrivances, alters the form of certain parts of the surface of the oil. Tho free portions of the surface then assume new forms depend ing on the equilibrium of surface-tension. In this way he has produced a great many of the forms of equilibrium of a liquid under the action of surface-tension alone, and com pared them with the results of mathematical investigation. He has also greatly facilitated the study of liquid films by showing how to form a liquid, the films of which will last for twelve or even for twenty-four hours. The debt which science owes to M. Plateau is not diminished by the fact that, while investigating these beautiful phenomena, he has never himself seen them. He lost his sight long ago in tho pursuit of science, and has ever since been obliged to depend on the eyes and the hands of others. M. Van der Mensbrugghe 3 has also devised a great number of beautiful illustrations of the phenomena of surface- tension, and has shown their connection with the experi ments of Mr Tomlinson on the figures formed by oils dropped on the clean surface of water. M. Dupr4 in his 5th, 6th, and 7th Memoirs on the Mechanical Theory of Heat (Ann. de C/timie et de Physique, 1866 to 1868) has done much towards applying the prin ciples of thermodynamics to capillary phenomena, and the experiments of his son are exceedingly ingenious and well devised, tracing the influence of surface tension in a great number of very different circumstances, and deducing from independent methods the numerical value of the sur face-tension. The experimental evidence which M Dupre has obtained bearing on the molecular structure of liquids must be very valuable, even if many of our present opinions on this subject should turn out to be erroneous. M. Quincke 4 has made a most elaborate series of experi ments on the tension of the surfaces separating one liquid from another and from air. M. Liidtge 5 has experimented on liquid films, and has shown how a film of a liquid of high surface-tension is re placed by a film of lower surface-tension. He has also ex perimented on the effects of the thickness of the film, and has come to the conclusion that the thinner a film is, the greater is its tension. This result, however, has been tested by M. Van der Mensbrugghe, who finds that the tension is the same for the same liquid whatever be the thickness, as long as the film does not burst. The phenomena of very thin liquid films deserve the most careful study, for it is in this way that we are most likely to obtain evidence by which we may test the theories of the molecular structure of liquids. Sir W. Thomson 6 has investigated the effect of the cur vature of the surface of a liquid on the thermal equilibrium between the liquid and the vapour in contact with it. He has also calculated the effect of surface-tension on the pro pagation of waves on the surface of a liquid, and has deter mined the minimum velocity of a wave, and the velocity of the wind when it is just sufficient to disturb the surface of still water. 7 THEORY OF CAPILLARY ACTION. When two different fluids are placed in contact, they may either diffuse into each other or remain separate. In 3 Mem. de VAcad. Roy. de Belgique, xxxvii. (1873). 4 Pogg. Ann., cxxxix. (1870), p. 1. b Pogg. Ann., cxxxix. (1870), p. 620. 6 Proceedings R. S., Edinburgh, February 7, 1870,
7 Philosophical Magazine, November, 1871.