Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/196

the ambiguous sign being determined according as the circular polarization as right-handed or left-handed.

Substituting in the above differential equations, we have

$c_{1}^{2}l^{2}\mp \mu .{\frac {2\pi }{\tau }}.l^{3}$,

or

$l={\frac {1}{c_{1}}}\pm {\frac {\pi \mu }{\tau c_{1}^{4}}}$.

Since 1/l denotes the velocity of propagation, it is evident that the reciprocals of the velocities of propagation of a right-handed and left-handed beam differ by the quantity

${\frac {2\pi \mu }{\tau c_{1}^{4}}}$;

from which it is easily shown that the angle through which the plane of polarization of a plane-polarized beam rotates in unit length of path is

${\frac {2\pi ^{2}\mu }{\tau ^{2}c_{1}^{4}}}$.

If we neglect the variation of c, with the period of the light, this expression satisfies Biot's law that the angle of rotation in unit length of path is proportional to the inverse square of the wave-length.

MacCullagh's investigation can be scarcely called a theory, for it amounts only to a reduction of the phenomena to empirical, though mathematical, laws; but it was on this foundation that later workers built the theory which is now accepted. ^[1]

1. The later developments of this theory will be discussed in a subsequent chapter; but mention may here be made of an attempt which was made in 1856 by Carl Neumann, then a very young man, to provide a rational basis for MacCullagh's equations. Neumann showed that the equations may be derived from the hypothesis that the relative displacement of one aethereal particle with respect to another acts on the latter according to the same law as an element of an electric current acts on a magnetic pole. Cf. the preface to C. Noumann's Die magnetische Drehung der Polarisationsebene des Lichtes, Halle, 1863.