Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/176

which shows that transverse waves are propagated with velocity $\textstyle {\sqrt {\mu /\rho }}$.

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 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

$\rho {\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=-{\frac {\partial }{\partial y}}(\mu {\text{ curl }}\mathbf {e} )_{z}+{\frac {\partial }{\partial z}}(\mu {\text{ curl }}\mathbf {e} )_{y}$

and as the quantities ρ, (μ curl e)_z and (μ curl e)_y, are continuous across the interface, the continuity of ∂²e_x/∂t² follows. Thus the only independent boundary-conditions in MacCullagh's theory are the continuity of the tangential components of e and of μ curl e.^[1] 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

$A\left({\frac {\partial e_{z}}{\partial y}}-{\frac {\partial e_{y}}{\partial z}}\right)^{2}+B\left({\frac {\partial e_{x}}{\partial z}}-{\frac {\partial e_{z}}{\partial x}}\right)^{2}+C\left({\frac {\partial e_{y}}{\partial x}}-{\frac {\partial e_{x}}{\partial y}}\right)^{2}$

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

1. MacCullagh's equations may readily be interpreted in the electro-magnetic theory of light: e corresponds to the magnetic force, μ curl e to the electric force, and curl e to the electric displacement.