156 The Aether as an Elastic Solid. 



which shows that transverse waves are propagated with velocity 



From the variational equation we may also determine the 

 boundary-conditions which must be satisfied at the interface 

 between two media ; these are, that the three components of e 

 are to be continuous across the interface, and that the two 

 components of p curl e parallel to the interface are also to be 

 continuous across it. One of these five conditions, namely, the 

 continuity of the normal component of e, is really dependent on 

 the other four ; for if we take the axis of x normal to the 

 interface, the equation of motion gives 



p a"? = ~ 3 ^ ( curl e) * + l* curl e) " / ' . | 



and as the quantities p, (n curl e) 2 , and (/n curl e) y are continuous 

 across the interface, the continuity of c> 2 e x /dt z follows. Thus the 

 only independent boundary-conditions in MacCullagh's theory 

 are the continuity of the tangential components of e and of 

 fj curl e.* It is easily seen that these are equivalent to the 

 boundary-conditions used in MacCullagh's earlier paper, namely, 

 the equation of vis viva and the continuity of the three 

 components of e : and thus the " rotationally elastic " aether of 

 this memoir furnishes a dynamical foundation for the memoir 

 of 1837. 



The extension to crystalline media is made by assuming the 

 potential energy per unit volume to have, when referred to the 

 principal axes, the form 



\dz dx J \c>x ty J 



where A, B, C denote three constants which determine the 

 optical behaviour of the medium : it is readily seen that the 

 wave-surface is Fresnel's, and that the plane of polarization 



* MacCullagh's equations may readily be interpreted in the electro -magnetic 

 theory of light : e corresponds to the magnetic force, p curl e to the electric force, 

 and curl e to the electric displacement. 



