THE ELECTRIC ANJI LUMINIFEROUS MEDIUM. 
729 
energy dejDends on the components of the strain of the medium, as it would do if the 
medium possessed the properties of an elastic solid. 
At any rate, MacCullagh assumes a purely rotational quadratic expression for 
the energy, which he reduces to its principal axes in the ordinary manner; and then 
he deduces from it in natural and easy sequence, without a hitch, or any forcing of 
constants, all the known laws of j^ropagation and reflexion for transparent isotropic 
and crystalline media. In common with Neumann, he cannot understand how with 
Feesnel the inertia in a crystal could be diflerent in different directions, or its 
elasticity isotropic; so he assumes the density of the mther to be the same in all 
media, hut its elasticity to be variable. The laws of crystalline reflexion are then 
established as below, and shown to be embraced in a single theorem relating either 
to his transversals or to his polar plane ; and the memoir ends with a remark “ which 
may be necessary to prevent any misconception as to the nature of the foundation on 
which ” the theory stands. “ Everything depends on the form of the function V; 
and we have seen that, when that form is properly assigned, tlie laws by which 
crystals act upon light are included in the general ec^uations of dynamics. This fact 
is fully proved by the foregoing investigations. But the reasoning which has been 
used to account for the form of the function is indirect, and cannot be regarded as 
sufficient, in a mechanical point of view. It is, however, the only kind of reasoning 
that we are able to employ, as the constitution of the luminiferous medium is 
entirely unknown.” 
MacCullaglis Optical Equations. 
14. Let the components of the linear displacement of the primordial medium be 
represented by (^, 77 , 1), and let {f, g, h.) represent the curl or vorticity of this 
displacement, i.e. 
dp ^ dp d^\ 
dy dz dz dx dx dy J 
so that this vector is equal to twice the absolute rotation of the element of volume. 
The elasticity being purely rotational, the potential energy per unit volume of the 
strained medium is represented by a cjuadratic function U of [f, g, h), so that 
W = |U dr 
where dj denotes an element of volume. The kinetic energy is 
^ ^ \\dt^ ^ dt^dty ^ 
The general variational equation of motion is 
Sf(T - Vil)dt = 0, 
5 A 
MBCCCXCIV.—A. 
