98 ON DEPARTURES FROM FRESNEL's LAWS OF REFLEXION [362
in which. K, p are the electric and magnetic constants for the first medium, Kl, fa for the second*. The relation between 6 and 6t is
Kifr : K iii, = sim* 6 : sin2^ .......................... (C)
It is evident that mere absence of refraction will not secure the evanescence of reflexion for both polarizations, unless we assume both ^ = //, and Kl — K. In the usual theory ^ is supposed equal to /JL in all cases. (A) then identifies itself with Fresnel's sine-formula, and (B) with the tangent-formula, and both vanish when K1 = K corresponding to no refraction. Further, (B) vanishes at the Brewsterian angle, even though there be refraction. A slight departure from these laws would easily be accounted for by a difference between /^ and /JL, such as in fact occurs in some degree (diamagnetism). But the effect of such a departure is not to interfere with the complete evanescence of (B), but merely to displace the angle at which it occurs from the Brewsterian value. If yu,1//u,= 1 + h, where h is small, calculation shows that the angle of complete polarization is changed by the amount
n being the refractive index. The failure of the diamond and dense glass to polarize completely at some angle of incidence is not to be explained in this way.
As I formerly suggested, the anomalies may perhaps be connected with the fact that one at least of the media is dispersive. A good deal depends upon the cause of the dispersion. In the case of a stretched string, vibrating transversely and endowed with a moderate amount of stiffness, the boundary conditions would certainly be such as would entail a reflexion in spite of equal velocity of wave-propagation. All optical dispersion is now supposed to be of the same nature as what used to be called anomalous dispersion, i.e. to be due to resonances lying beyond the visible range. In the simplest form of this theory, as given by Maxwell f and Sellmeier, the resonating bodies take their motion from those parts of the sether with which they are directly connected, but they do not influence one another. In such a case the boundary conditions involve merely the continuity of the displacement and its first derivative, and no complication ensues. When there is no refraction, there is also no reflexion. By introducing a mutual reaction between the resonators, and probably in other ways, it would be possible to modify the situation in such a manner that the boundary conditions would involve higher derivatives, as in the case of the stiff string, and thus to allow reflexion in spite of equality of wave- velocities for a given ray.
* " On the Electromagnetic Theory of Light," Phil. Mag. Vol. xii. p. 81 (1881) ; Scientific Papers, Vol. i. p. 521.
t Cambridge Calendar for 1869. See Phil. Mag. Vol. XLVIII. p. 151 (1899); Scientific Papers, Vol. iv. p. 413.>\