| ψ(z) = ½ a(a² − z²) = 1⁄3(a³ − z³) | } (44) |
| from z = 0 to z = a, | |
| ψ(z) = 0 from z = a to z = ∞, |
Equations (37) now give
| K0 = | 2π | ∫∞0z3 dz = | πa4 | , (45) |
| 3 | 6 |
| T0 = | π | ∫∞0z4 dz = | πa5 | . (46) |
| 8 | 40 |
The numerical results differ from those of Young, who finds that “the contractile force is one-third of the whole cohesive force of a stratum of particles, equal in thickness to the interval to which the primitive equable cohesion extends,” viz. T = 1⁄3 aK; whereas according to the above calculation T = 3⁄20 aK. The discrepancy seems to depend upon Young having treated the attractive force as operative in one direction only. For further calculations on Laplace’s principles, see Rayleigh, Phil. Mag., Oct. Dec. 1890, or Scientific Papers, vol. iii. p. 397.]
On Surface-Tension
Definition.—The tension of a liquid surface across any line drawn on the surface is normal to the line, and is the same for all directions of the line, and is measured by the force across an element of the line divided by the length of that element.
Experimental Laws of Surface-Tension.—1. For any given liquid surface, as the surface which separates water from air, or oil from water, the surface-tension is the same at every point of the surface and in every direction. It is also practically independent of the curvature of the surface, although it appears from the mathematical theory that there is a slight increase of tension where the mean curvature of the surface is concave, and a slight diminution where it is convex. The amount of this increase and diminution is too small to be directly measured, though it has a certain theoretical importance in the explanation of the equilibrium of the superficial layer of the liquid where it is inclined to the horizon.
2. The surface-tension diminishes as the temperature rises, and when the temperature reaches that of the critical point at which the distinction between the liquid and its vapour ceases, it has been observed by Andrews that the capillary action also vanishes. The early writers on capillary action supposed that the diminution of capillary action was due simply to the change of density corresponding to the rise of temperature, and, therefore, assuming the surface-tension to vary as the square of the density, they deduced its variations from the observed dilatation of the liquid by heat. This assumption, however, does not appear to be verified by the experiments of Brunner and Wolff on the rise of water in tubes at different temperatures.
3. The tension of the surface separating two liquids which do not mix cannot be deduced by any known method from the tensions of the surfaces of the liquids when separately in contact with air.