Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/200

Χwhere

${\frac {a}{c}}=-\left({\frac {\rho }{n}}\right)^{\frac {1}{2}}\cos i$, $\qquad {\frac {b}{c}}=-\left({\frac {\rho }{n}}\right)^{\frac {1}{2}}\sin i$;

here i denotes the angle of incidence, ρ the inertia of the aether, and n its rigidity.

Let the reflected light be a function of the argument

$a_{1}x+by+ct$,

where, in order to secure continuity at the boundary, b and c must have the same values as before. Since Green's formulae are to be still applicable, we must have

${\frac {a_{1}}{b}}=\cot r$,

where $\sin i=\mu \sin r$, but μ has now a complex value. This equation may be written in the form

$a_{1}^{2}+b^{2}={\frac {\mu ^{2}\rho c^{2}}{n}}$.

Let the complex value of μ² be written

$\mu ^{2}={\frac {\rho _{1}}{\rho }}-A{\sqrt {-1}}$,

the real part being written ρ₁/ρ in order to exhibit the analogy with Green's theory of transparent media: then we have

$a_{1}^{2}+b^{2}={\frac {\rho _{1}}{n}}c^{2}-{\frac {\rho c^{2}}{n}}A{\sqrt {-1}}$.

But an equation of this kind must (as in Green's theory) represent the condition to be satisfied in order that the quantity

$e^{(a_{1}x+by+ct){\sqrt {-1}}}$

may satisfy the differential equation of motion of the aether; from which we see that the equation of motion of the aether in the metallic medium is probably of the form

$\rho _{1}{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}+\rho cA{\frac {\partial e_{z}}{\partial t}}=n\left({\frac {\partial ^{2}e_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}e_{z}}{\partial y^{2}}}\right)$.

This equation of motion differs from that of a Greenian