If denote the circumference of a circle that has its radius equal to unit, and r the radius of the ca pillary tube, then (H'--41-1) x rq will be the weight cif the elevated column of fluid within the tube, and rir will be its bulk. Conceive a plane which touches the curve surface of the column at its lowest point, and let q be the height of that point above the level on the outside of the tube, then the ele vated column will consist of a cylinder equal to and a small meniscus above the cylinder ; so that, in very small tubes, the cylinder may be taken for the whole bulk of the column; wherefore, by equating the two expressions of the same bulk, we get x which proves, that, in small tubes of the same mat ter, the elevations are reciprocally proportional to the radii, or diameters of the tubes.
And because te is the same in all cases, when plates and tubes of the same matter act on the same fluid, if we equate the values of it taken from the last expression, and from the expression formerly ob tained for two plates, we shall get HxQr_-_rxq; and this shows, that a fluid will rise between two plates, to the same height it would do in a tube of the same matter, having its radius equal to the dis tance of the plates.
The deductions that have now been drawn from the principle of a corpuscular attraction evanescent at all sensible distances, are equivalent to the account of capillary action founded on the hypothesis of Dr Jurin. Whatever may be thought of the physical principle advanced by this philosopher, it must be allowed, that his theory agrees well with observa tion ; and it cannot be denied, that he has, with great sagacity, inferred from his experiments the true place in which the capillary force resides. But it is impossible to accede to his opinion, that, when a capillary tube of glass is immersed in water, the water within the tube is attracted upward by a nar row ring of glass immediately above the surface of the liquid. If the glass attract the water, the at traction must be perpendicular to the surface of the glass ; the force acting on the fluid cannot be verti cal, it must be horizontal ; and if we would rease4 strictly, the proper inference must be, that an at traction between the glass and the water is alone in sufficient to account for capillary action. In order to explain the phenomena, it is necessary to attend to the remark of Professor Leslie, founded on the properties essential to fluidity, namely, that a fluid cannot be attracted horizontally by a solid body, without having a vertical force communicated to it. It is certainly not a little surprising, that an ob servation made in 1802, so well calculated to remove all the difficulties of the theory, should have passed entirely unnoticed, although, since that period, the subject has engaged the attention of the first philo sophers of the age.
In what goes before, it has been shown, that, in many cases, the height to which a fluid will rise may be found with considerable exactness, by comparing the bulk as determined by the magnitude of the ca pillary force with the same bulk deduced from the figure which the displaced fluid is constrained to assume ; but a rigorous investigation of all the cir cumstances attending the capillary phenomena re quires further, that we know the nature of the curve assumed by that part of the fluid's surface, which is free to obey the impulse of all the forces that act upon it ; and it is to this branch of the subject that we are now to proceed.
8. Resuming the first and simplest case of a single plate immersed in a fluid, which rises upon its surface in a concave ring, let a vertical plane PL (fig. 4.), pa. rallel to the plate, and at a distance from its surface greater than the range of the corpuscular force, be drawn to intersect the curve, then the part of the ring cut off, being without the sphere of the plate's attraction, must be supported by the force with which it is attracted by the fluid between the plate and the plane. Now, all the fluid below the super ficial stratum is in equilibrium with regard to the corpuscular forces to which it is subjected ; and hence it is the attraction of the fluid between the plate and the plane upon the superficial stratum, which sup ports the part of the ring below the plane, in the same manner that the attraction of the plate upon the same stratum supports the whole ring. All the fluid in the vessel being supposed in equilibrium, we may. conceive that the portion of it between the plate and the plane is converted into a solid without any other change of its properties ; then, if we consider that part of the superficial canal which lies between the vertical plane and the level surface of the fluid, the upper end of it will be pressed against the imagi nary solid by the attraction of an obtuse-angled wedge of the fluid, while the pressure upon all the other vertical sides is only equal to the attraction of a right-angled wedge ; and the difference of these forces remaining unbalanced, generates the force which tends upward, and supports the weight of the part of the ring situated below the point of its ac tion.
It is now necessary to determine the attractive force of a portion of a fluid, in the shape of a wedge, con tained in any proposed angle. Suppose that a fluid mass bounded by the plane AB (fig. 6.) is divided by the plane PQ; and let it be required to find the force with which the attraction of the particles con tained in each of the wedges APQ and BPQ, will cause a small drop placed at P to press upon the plane AB. Draw PN and PG to bisect the angles