Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/160

This page has been proofread, but needs to be validated.

140

The Aether as an Elastic Solid.

two constants, k and n, appear instead of one. The reason for this is that a body constituted from point-centres of force in Navier's fashion has its moduli of rigidity and compression connected by the relation^[1]

$k={\frac {5}{3}}n$ .

Actual bodies do not necessarily obey this condition; e.g. for india-rubber, k is much larger than ${\frac {5}{3}}n$ ;^[2] and there seems to be no reason why we should impose it on the aether.

In the same year Poisson^[3] succeeded in solving the differential equation which had thus been shown to determine the wave-motions possible in an elastic solid. The solution, which is both simple and elegant, may be derived as follows:—Let the displacement vector e be resolved into two components, of which one c is circuital, or satisfies the condition

div c = 0,

while the other b is irrotational, or satisfies the condition

curl b = 0.

The equation takes the form

$\rho {\frac {\partial ^{2}}{\partial t^{2}}}(\mathbf {b} +\mathbf {c} )-n\nabla ^{2}\mathbf {c} -\left(k+{\frac {4}{3n}}\right)\nabla ^{2}\mathbf {b} =0$ .

↑ In order to construct a body whose elastic properties are not limited by this equation, William John Macquorn Rankine (b.1820, d. 1872) considered a continuous fluid in which a number of point-centres of force are situated: the fluid is supposed to be partially condensed round these centres, the elastic atmosphere of each nucleus being retained round it by attraction. An additional volume-elasticity due to the fluid is thus acquired; and no relation between k and n is now necessary. Cf. Rankine's Miscellaneous Scientific Papers, pp. 81 994. Sir William Thomson (Lord Kelvin), in 1889, formed a solid not obeying Navier's condition by using pairs of dissimilar atoms. Of. Thomson's Papers, iii, p. 396. Cf. also Baltimore Lectures, pp. 123 sqq.
↑ It may, however, be objected that india-rubber and other bodies which fail to fulfil Navier's relation are not true solids. On this historic controversy, cf. Todhunter and Pearson's History of Elasticity, i, p. 496.
↑ Mém. de l'Acad., viii (1828), p. 623. Poisson takes the equation in the restricted form given by Navier; but this does not affect the question of wave-propagation.

[1] In order to construct a body whose elastic properties are not limited by this equation, William John Macquorn Rankine (b.1820, d. 1872) considered a continuous fluid in which a number of point-centres of force are situated: the fluid is supposed to be partially condensed round these centres, the elastic atmosphere of each nucleus being retained round it by attraction. An additional volume-elasticity due to the fluid is thus acquired; and no relation between k and n is now necessary. Cf. Rankine's Miscellaneous Scientific Papers, pp. 81 994. Sir William Thomson (Lord Kelvin), in 1889, formed a solid not obeying Navier's condition by using pairs of dissimilar atoms. Of. Thomson's Papers, iii, p. 396. Cf. also Baltimore Lectures, pp. 123 sqq.

[2] It may, however, be objected that india-rubber and other bodies which fail to fulfil Navier's relation are not true solids. On this historic controversy, cf. Todhunter and Pearson's History of Elasticity, i, p. 496.

[3] Mém. de l'Acad., viii (1828), p. 623. Poisson takes the equation in the restricted form given by Navier; but this does not affect the question of wave-propagation.

[1]

[2]

[3]