1 72 The Aether as an Elastic Solid. 



The variational equation of motion is 



pi ^r/ de x + p z % de y + p z 2 $e z [ dx dy dz 



where p denotes an undetermined function of (x, y, z) : the term 

 in p being introduced on account of the kinematical constraint 

 expressed by the equation 



div e = 0. 



The equations of motion which result from this variational 

 equation are 



< h w = - +nV * e " '""' 



and two similar equations. It is evident that p resembles a 

 hydrostatic pressure. 



Substituting in these equations the analytical expression 

 for a plane wave, we readily find that the velocity F of the 

 wave is connected with the direction-cosines (X, ^t, z/) of its 

 normal by the equation 



A 2 u* v z 



n- PI V* r n-ptV" + n- p z V* = ' 



When this is compared with Fresnel's relation between the 

 velocity and direction of a wave, it is seen that the new formula 

 differs from his only in having the reciprocal of the velocity in 

 place of the velocity. About 1867 Stokes carried out a series 

 of experiments in order to determine which of the two theories 

 was most nearly conformable to the facts : he found the con- 

 struction of Huygens and Fresnel to be decidedly the more 

 correct, the difference between the results of it and the rival 

 construction being about 100 times the probable error of 

 observation.* 



* Proc. R. S., June, 1872. After these experiments Stokes gave it as his opinion 

 (Phil. Mag. xli (1871), p. 521) that the true theory of crystal-optics was yet to be 

 found. On the accuracy of Fresnel's construction cf. Glazebrook, Phil. Trans, 

 clxxi (1879) p. 421, and Hastings, Am. Journ. Sci. (3) xxxv (1887) p. 60. 



