Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/234

 In the year following Faraday's discovery, Airy suggested a way of representing the effect analytically; as might have been expected, this was by modifying the equations which had been already introduced by MacCullagh for the case of naturally active bodies. In MacCullagh's equations

the terms $$\tfrac{\partial^3 Z}{\partial x^3}$$ and $$\tfrac{\partial^3 Y}{\partial x^3}$$ change sign with x, so that the rotation of the plane of polarization is always right-handed or always left-handed with respect to the direction of the beam. This is the case in naturally-active bodies; but the rotation due to a magnetic field is in the same absolute direction whichever way the light is travelling, so that the derivations with respect to x must be of even order. Airy proposed the equations

where μ denotes a constant, proportional to the strength of the magnetic field which is used to produce the effect. He remarked, however, that instead of taking $$\mu\tfrac{\partial Z}{\partial t}$$ and $$\mu\tfrac{\partial Y}{\partial t}$$ as the additional terms, it would be possible to take $$\mu\tfrac{\partial^3 Z}{\partial t^3}$$ and $$\mu\tfrac{\partial^3 Y}{\partial t^3}$$, or $$\mu\tfrac{\partial^3 Z}{\partial x^2 \partial t}$$ and $$\mu\tfrac{\partial^3 Z}{\partial x^2 \partial t}$$, or any other derivates in which the number of differentiations is odd with respect to t and even with respect to x. It may, in fact, be shown by the method previously applied to MacCullagh's formulae that, if the equations are where (r + s) is an odd number, the angle through which the