Some philosophers account for capillary action, by means of attractions between the plate and the fluid, which are supposed to take place, partly at the sur face of the fluid, and partly at the bottom of the plate. Laplace, in particular, has grounded his se cond, or more popular theory, entirely on such at tractions. He observes, that the part of the plate's vertical plane, immersed in the water, attracts the fluid in contact with it as much upward as down ward, and therefore has no effect in causing either an elevation or a depression ; but the part above the water attracts a thin film in contact with the plate upward; and the whole vertical side of the plate likewise attracts in the same direction the fluid be low it, and situated in its prolongation. According to Laplace, it is the united effect of these two at. tractions which supports the suspended ring. The whole of this reasoning appears to us gratuitous. No part of the fluid is attracted by the solid matter in a vertical direction, but in a direction perpendicular to the plate's surface. The immersed part of the solid presents a continuous surface to the fluid, at tracting it with the same intensity at every point ; whereas Laplace neglects the action of the plate's horizontal boundary, and seems to suppose that the attractive energy of the solid matter resides only in the vertical sides. We have endeavoured to prove, that the thin film, or coating of fluid, which covers the part of the solid immersed below the superficial stratum, is every where in a state of equilibrium and of equal compression by the attractions which act upon it. There therefore, no force produced at the bottom of the plate by the attraction between the solid matter and the particles of the fluid, which can contribute to support the weight of the ring raised above the level.
We proceed now to consider the action of the plate upon the superficial stratum. Trace a canal at right angles to the plate, of the same depth with the superficial film, and having its horizontal width equal to unit, and continue the canal till it termi nate in a vertical plane PS, parallel to the plate. Let n be the small portion of the canal within the sphere of the plate's attraction, and suppose the ca nal to be divided in its whole length into the pa rallelopipeds in, m, m, &c., each equal to n. It is plain that the attractions of the fluid below the ca nal, and on the two sides of it, have no tendency to impel it in any direction, nor to impede the motion of the fluid along it. The canal is also in equili brium with regard to gravity, since by the hypothesis it is horizontal. The rectangular wedge of fluid be yond the plane PS will attract the small parallelopiped contiguous to it with a force proportional to /K; be cause K denotes the attractive force of two rectangular wedges, § 5 ; and the same parallelopiped will also be attracted with an equal force in the opposite direction by the one next to it. In like manner, every parallel opiped in the canal is attracted with equal forces by those contiguous to it on opposite sides, except the one in contact with the plate, which is attracted in the direction of the canal with the force' K, and towards the plate, with the force K', depending upon the intensity of the plate's attraction for the fluid.
Now, if K' be just equal to IK, which will happen whenthe intensity of the plate's attraction is halt that of the ffuid, the parallelopiped n will be situated with regard to the forces that act upon it, similarly to the others in the canal; in this case, therefore, the in sertion of the plate will not disturb the equilibrium of the fluid, the surface of which will remain hori zontal.
If K' be greater than IK, it may be resolved in to two parts, iK —1K), of which one will counterbalance the opposite force, and reduce the canal to equilibrium ; and the other part, K'--IK, will act only upon the parallelopiped n, and will compress it upon the surface of the plate. The compression will produce a lateral force proportional to H'—.11-1, which urges the small fluid mass to spread itself towards every side ; and, as this force is unopposed vertically upward, the equilibrium of the fluid will be disturbed ; the superficial film will ascend all round the plate, and, by means of the force of cohesion, will carry with it a portion of the fluid till the suspended weight is sufficient to coun terbalance the force acting upward.
When K' is less than the parallelopiped in contact with the plate will be more attracted in the direction of the canal than towards the plate. When this happens the fluid is depressed below the level by capillary action; but we shall leave this case to be afterwards considered, and at present confine our attention to the former case, when the fluid is ele vated above the level.
7. When the immersion of the plate causes an ele vation, the fluid will assume the form of a concave ring as KLM (Plate LXXX. fig. 4.). If we suppose a su perficial canal divided into parallelopipeds as before, we may prove, by like reasoning, that the attraction of the solid matter has no tendency to disturb the equili brium of the fluid except by the lateral force which it communicates to the small parallelopiped in contact with it. And since the attractive force of the plate upon the particles of the fluid depends only upon their perpendicular distance from its surface, it rea dily follows that the lateral force will undergo no variation, but will remain constantly equal to both during the rising of the ring, and when it has attained the greatest elevation. The reciprocal at traction of all the fluid in the vessel, likewise pro duces pressures that are propagated inward from the surface of the fluid, and from the sides and bot tom of the vessel, § 3 ; but these forces cannot be in equilibrium with the weight of the ring and the disturbing force arising from the plate's attraction. For the former forces have no tendency to move the centre of gravity of the whole mass, whereas the latter tend to produce motion in that point each in its own direction. In the case of equilibrium, therefore, the vertical force arising from the plate's attraction must be equal to the weight of the sus pended ring ; or, which is the same thing, will express the weight of a portion of the ring in every unit of the horizontal extent.