This expression would evidently denote the pressure upon the surface abdc, if the fluid below the section were impelled towards the plate by a piston exact ly fitted to the orifice of the tube. But there is no difference between the action of such a piston and that of the thin elementary section when urged by attraction with equal force in the same direction. The total force acting laterally in the length X, is, therefore, equal to the fluent generated while a increases from o to be infinitely great. Hence, if we put 114;* (a).ada ; then H' will be the measure of the lateral force in the length equal to unit.
It is obvious, that the direct attraction between two portions of a fluid, as well as that between a solid and a fluid, is attended with a lateral pressure. If we denote by H, what H' becomes when the matter of the plate attracts the fluid with the same intensity that the fluid attracts its own particles, then will be the measure of the lateral force arising from the direct attraction of the fluid, and it will have the same relation to K that H' has to K'.
The lateral force is always very small when com pared with the direct pressure. For the function 11(a) has a conceivable value only when a is so small as to be imperceptible to the senses ; in such cir cumstances, the product Ni(a) x a is very small when compared with v(a ) ; and, consequently, H'= is considerable in respect of The smallness of the lateral, in comparison of the direct, pressure, arises from this, that every elemen tary part of the latter is estimated on the same finite area, while the simultaneous element of the former is confined to a space incomparably less. These two pressures resemble the power in the hydro static paradox and the effect which it produces. In both cases we have a small pressure applied to a surface extremely minute, in equilibrium with a great pressure distributed over a comparatively large area.
When a piece of glass is partially plunged in water in a vertical direction, the thin film which is attract ed by the immersed surface endeavours to spread itself on the glass with an effort more or less in pro portion to the compressive force. Below the sur face of the water, the lateral actions of the parts in contact mutually counteract one another ; but at the surface, the expansive force meets with no opposi tion. The film will, therefore, be pushed above the general level, and as it acts by cohesion on the con tiguous fluid, it will draw upward a portion of it, and form a ring surrounding the immersed part of the glass. The small fluid mass on which the glass ex
erts its attraction performs the office of a machine, which changes a horizontal force into one having a vertical direction. In the mechanical properties of a fluid, we thus have a principle adequate to ac count for what we observe in capillary action. But although the general view here given of the cause of the capillary phenomena is so far satisfactory, a great deal of discussion is still necessary, in order to deduce from it a clear explanation of the laws observed in the appearances that take place in dif ferent circumstances.
The idea of accounting for capillary action by means of the lateral force produced by the direct attraction of a solid body upun a fluid, is due to Pro fessor Leslie, a philosopher to whom physical science is indebted for more than one discovery. It is de veloped and applied, to explain some of the princi pal phenomena, in a short dissertation published, in 1802, in the Philosophical Magazine. This disser tation is written with tee same ability that charac terizes all the productions of the author, and nothing more was necessary than to pursue the observation he had made, in order to obtain a complete theory of this branch of natural philosophy. It happens that, in this instance, the views of the philosopher are confirmed by the most abstruse and refined ma thematical investigation. The formula found by La place, for the attractive force of a fluid bounded by a curve surface, consists of two parts, one of which is the same for all surfaces, and the other varies with the curvature in each particular case. The first of these terms is the attractive force of an indefinite mass of the fluid bounded by a plane. The other term, which depends upon the curvature, is compo sed of a constant quantity multiplied into half the sum of the reciprocals of the radii of the circles, which have the same curvature with any two sec tions of the curve surface made by planes, perpen dicular to one another, and to the curve surface ; and, on examination, this constant quantity will be found to coincide with the measure of the lateral ten dency of the fluid caused by the direct action of the first force. Thus it appears, that the two quantities which enter into the formula of Laplace are no other than the measures of the two kinds of force which we have been considering ; the one denoting the direct pressure caused by the attraction of a fluid mass bounded by a plane, and the other signifying the derivative force acting laterally, which is a ne cessary consequence of the direct pressure. In a subsequent part of this article, what has now been advanced will be proved, by deducing the formula of Laplace in a direct and satisfactory manner from the two kinds of force, with the consideration of which we have been occupied.