The subject was next taken up by Pierre Simon Laplace (Mecanique celeste, supplement to the tenth book, pub. in 1806). His results are in many respects identical with those of Young, but his methods of arriving at them are very different, being conducted entirely by mathematical calculations. For those who wish to study the molecular constitution of bodies it is necessary to study the effect of forces which are sensible only at insensible distances ; and Laplace has furnished us with an example of the method of this study which has never been surpassed. He found for the pressure at a point in the interior of the fluid an expres sion of the form where K is a constant pressure, probably very large, which, however, does not influence capillary phenomena, and therefore cannot be determined from observation of such phenomena; H is another constant on which all capillary phenomena depend; and R and R' are the radii of curvature of any two normal sections of the surface at right angles to each other.
The next great step in the treatment of the subject was made by C. F. Gauss (Principia generalia Theoriae Figurae Fluidorum in state Aequilibrii, Gottingen, 1830, or Werke, v. 29, Gottingen, 1867). The principle which he adopted is that of virtual velo cities, a principle which under his hands was gradually trans forming itself into what is now known as the principle of the conservation of energy. Instead of calculating the direction and magnitude of the resultant force on each particle arising from the action of neighbouring particles, he formed a single expression which is the aggregate of all the potentials arising from the mutual action between pairs of particles. This expression has been called the force-function. With its sign reversed it is now called the potential energy of the system. It consists of three parts, the first depending on the action of gravity, the second on the mutual action between the particles of the fluid, and the third on the action between the particles of the fluid and the particles of a solid or fluid in contact with it.
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 estab lished by Laplace, but the conditions of the angle of contact of this surface, with the surface of a solid.
In 1831 Simeon Denis Poisson published his Nouvelle Theorie de l'action capillaire. He maintained that there is a rapid variation of density near the surface of a liquid, and he gave very strong reasons, which have been only strengthened by subsequent dis coveries, for believing that this is the case.
The result, however, of Poisson's investigation is practically 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 de pends 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 now 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. See Enrico Betti, Teoria della capii larita: Nuovo Cimento (1867); a memoir by M. Stahl, "Ueber einige Punckte in der Theorie der Capillarerscheinungen," Pogg. Ann. cxxxix. p. 239 (1870) ; and J. D. Van der Waals' Over de Continuiteit van den Gasen Vloeistoftoestand. A good account of the subject from a mathematical point of view will be found in James Challis's "Report on the Theory of Capillary Attraction," Brit. Assn. Report, iv. p. 235 J. A. F. Plateau (Statique experimentale et theorique des liquides, 1873), who made elaborate study of the phenomena of surface tension, adopted the following method of getting rid of the effects of gravity. He formed a mixture of alcohol and water of the same density as olive oil, and then introduced 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, discs and other contrivances, altered the form of certain parts of the surface of the oil. The free portions of the surface then assume new forms depending on the equilibrium of surface tension. In this way he produced a great many of the forms of equilibrium of a liquid under the action of surface tension alone, and compared them with the results of mathe matical investigation. The debt which science owes to Plateau is not diminished by the fact that, while investigating these beauti ful phenomena, he never himself saw them, having lost his sight in about 1840.
G. L. van der Mensbrugghe (Mem. de l' Acad. Roy. de Belgique, xxxvii., 1873) devised a great number of beautiful illustrations of the phenomena of surface tension, and showed their connection with the experiments of Charles Tomlinson on the figures formed by oils dropped on the clean surface of water.
Athanase Dupre in his 5th, 6th and 7th Memoirs on the Mechanical Theory of Heat (Ann. de Chimie et de Physique, 1866-68) applied the principles of thermodynamics to capillary phenomena, and the experiments of his son Paul were 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 surface tension. The experimental evidence which Dupre obtained bearing on the molecular structure of liquids must be very valu able, even if our present opinions on this subject should turn out to require modification.