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* 



Actual bodies do not necessarily obey this condition; e.g. 



for india-rubber, k is much larger than - n ;f and there seems to 



o 



be no reason why we should impose it on the aether. 



In the same year PoissonJ succeeded in solving the diffe- 

 rential 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 



+ 5 V b = ' 



o Tlj 



* 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 con- 

 tinuous 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 sqq. 



Sir "William Thomson (Lord Kelvin), in 1889, formed a solid not obeying 

 Navier's condition by using pairs of dissimilar atoms. Cf. Thomson's Papers, 

 iii, p. 395. Cf. also Baltimore Lectures, pp. 123 sqq. 



t 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. 



J Mem. de 1'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. 



