By means of the methods for calculating the attraction of spheroids, Ire determines the action of a mass of fluid terminated by a spherical surface, concave or convex, up on a column of fluid contained in an infinitely narrow canal, directed towards the centre of this surface. By this action La Place means the pressure which the fluid contained in the canal would exercise, in virtue of the attraction of its entire mass upon a plane base situated in the interior of the canal, and perpendicular to its sides, at any sensible distance from the surface, this base being taken for unity. He then shews that this action is smaller when the surface is concave than when it is plane, and greater when the sur face is convex. The analytical expression of :this action is composed of two terms : the first of these terms, which is much greater than the second, expresses the action of the mass terminated by a plane surface,* and the second term expresses the part of the action due to the sphericity of the surface, or, in other words, the action of the meniscus comprehended between this surface and the plane which touches it. This action is either additive to the preceding, or subtractive from it, acording as the surface is convex or concave. It is reciprocally proportional to the radius of the spherical surface ; for the smaller that this radius is, the meniscus is the nearer to the point of contact.
From these results relative to bodies terminated by sen sible segments of a spherical surface, La Place deduces this general theorem. 6, In all the laws which render the attraction insensible at sensible distances, the action of a body terminated by a curve surface upon an interior canal infinitely narrow, perpendicular to this surface in any point, is equal to the half sum of the actions upon the same canal of two spheres, which have for their radii the greatest and the smallest of the radii of the osculating circle of the sur face at this point." By means of this theorem, and the laws of hydrostatics, La Place has determined the figure which a mass of fluid ought to take when acted upon by gravity, or contained in a vessel of a given figure. The nature of the surface is expressed by an equation of partial differences of the se cond order, which cannot be integrated by any known method. If the figure of the surface is one of revolution, the equation is reduced to one of ordinary differences, and is capable of being integrated by approximation, when the surface is very small. La Place next shews, that a very narrow tube approaches the more to that of a spherical seg ment as the diameter of the tube becomes smaller. If these segments are similar in different tubes of the same substance, the radii of their surfaces will be inversely as the diameter of the tubes. This similarity of the spherical
segments will appear evident, if we consider that the dis tance at which the action of the tube ceases to be sensible, is imperceptible ; so that if, by means of a very powerful microscope, this distance should be found equal to a mil limetre, it is probable that the same magnifying power would give to the diameter of the tube an apparent dia meter of several metres. The surface of the tube may therefore be considered as very nearly plane, in a radius equal to that of the sphere of sensible activity ; the fluid in this interval will therefore descend, or rise from this sur face, very nearly as if it were plane. Beyond this the fluid being subjected only to the action of gravity, and the mu tual action of its own particles, the surface will be very nearly that of a spherical segment, of which the extreme planes being those of the fluid surface, at the limits of ti c sphere of the sensible activity of the tube, will be very nearly in different tubes equally inclined to their sides. Hence it follows that all the segments will be similar.
The approximation of these results gives the true cause of the ascent or descent of fluids in capillary tubes in the inverse ratio of their diameter. If in the axis of a glass tube we conceive a canal infinitely narrow, which bends round like the tube ABEDC in Fig. 7. the action of the water in the tube of this narrow canal, will be less on account of the concavity of its surface, than the action of water in the vessel on the same canal. The fluid will therefore rise in the tube to compensate for this difference of action ; and as the concavity is inversely proportional to the diameter of the tube, the height of the fluid will be also inversely proportional to that diameter. If the sur face of the interior fluid is convex, which is the case with mercury in a glass tube, the action of this fluid on the ca nal will be greater than that of the fluid in the vessel, and therefore the fluid will descend in the tube in the ratio of their difference, and consequently in the inverse ratio of the diameter of the tube.
In this manner of viewing the subject, the attraction of capillary tubes has no influence upon the ascent or depres sion of the fluids which they contain, but in determining the inclination of the first planes of the surface of the in terior fluid extremely near the sides of the tube, and up on this inclination depends the concavity or convexity of the surface, and the length of its radius. The friction of the fluid against the sides of the tube may augment or di minish a little the curvature of its surface, of which we see frequent examples in the barometer. In this case the capillary effects will increase or diminish in the same ratio.