Thomson, moreover, investigated[1] the ponderomotive forces which act between two solid bodies immersed in a fluid, when one of the bodies is constrained to perform small oscillations. If, for example, a small sphere immersed in an incompressible fluid is compelled to oscillate along the line which joins its centre to that of a much larger sphere, which is free, the free sphere will be attracted if it is denser than the fluid; while if it is less dense than the fluid, it will be repelled or attracted according as the ratio of its distance from the vibrator to its radius is greater or less than a certain quantity depending on the ratio of its density to the density of the fluid. Systems of this kind were afterwards extensively investigated by C. A. Bjerknes.[2] Bjerknes showed that two spheres which are immersed in an incompressible fluid, and which pulsate (i.e., change in volume) regularly, exert on each other (by the mediation of the fluid) an attraction, determined by the inverse square law, if the pulsations are concordant; and exert on each other a repulsion, determined likewise by the inverse square law, if the phases of the pulsations differ by half a period. It is necessary to suppose that the medium is incompressible, so that all pulsations are propagated instantaneously: otherwise attractions would change to repulsions and vice versa at distances greater than a quarter wave-length.[3] If the spheres, instead of pulsating, oscillate to and fro in straight lines about their mean positions, the forces between them are proportional in magnitude and the same in direction, but
- ↑ Phil. Mag. xli (1870), p. 427.
- ↑ Repertorium d. Mathematik von Konisberger und Zeuner (1876), p. 268. Göttinger Nachrichten, 1876, p. 245. Comptes Rendus, lxxxiv (1877), p. 1377. Cf. Nature, xxiv (1881), p. 360.
- ↑ On the mathematical theory of the force between two pulsating spheres in a fluid, cf. W. M. Hicks, Proc. Camb. Phil. Soc. iii (1879), p. 276: iv (1880), p. 29.
mutual distance l than when sinks of the same strengths are at infinite distance apart by an amount 4πρmm′/l. Since, in the case of the tubes, the quantities m correspond to the fluxes of fluid, this expression corresponds to the Lagrangian form of the kinetic energy: and therefore the force tending to increase the coordinate x of one of the sinks is (∂/∂x) (4πρmm′/l). Whence it is seen that the like ends of two tubes attract, and the unlike ends repel, according to the inverse square law.
