and since this must be the kinetic energy of the elastic medium, the density of the latter must be taken equal to g2, so that it must be the same in all substances.
It may further be asked what value we have to assign to the potential energy in the model, which must correspond to the electric energy in the electromagnetic field. Now, on account of (11) and (19), we can satisfy the equations (12) by putting Dx = gc (∂ζ/∂y − ∂η/∂z), &c., so that the electric energy (17) per unit of volume becomes
| ½ g2c2 { | 1 | ( | ∂ζ | − | ∂η | )2 + | 1 | ( | ∂ξ | − | ∂ζ | )2 + | 1 | ( | ∂η | − | ∂ξ | )2 }. |
| ε1 | ∂y | ∂z | ε2 | ∂z | ∂x | ε3 | ∂x | ∂y |
This, therefore, must be the potential energy in the model.
It may be shown, indeed, that, if the aether has a uniform constant density, and is so constituted that in any system, whether homogeneous or not, its potential energy per unit of volume can be represented by an expression of the form
| ½ { L ( | ∂ζ | − | ∂η | )2 + M ( | ∂ξ | − | ∂ζ | )2 + N ( | ∂η | − | ∂ξ | )2 }, |
| ∂y | ∂z | ∂z | ∂x | ∂x | ∂y |
(20)
where L, M, N are coefficients depending on the physical properties of the substance considered, the equations of motion will exactly correspond to the equations of the electromagnetic field.
18. Theories of Neumann, Green, and MacCullagh.—A theory of light in which the elastic aether has a uniform density, and in which the vibrations are supposed to be parallel to the plane of polarization, was developed by Franz Ernst Neumann,[27] who gave the first deduction of the formulas for crystalline reflection. Like Fresnel, he was, however, obliged to introduce some illegitimate assumptions and simplifications. Here again Green indicated a more rigorous treatment.
By specializing the formula for the potential energy of an anisotropic body he arrives at an expression which, if some of his coefficients are made to vanish and if the medium is supposed to be incompressible, differs from (20) only by the additional terms