256 REPORT—1862. 
berg’s indices* for the line D, I find that the velocities of propagation of the 
two polarized waves, according to the theory of Cauchy and Neumann, differ 
from those resulting from the theory of Fresnel only in the tenth place of 
decimals, the velocity in air being taken as unity. Such a difference as this 
would of course be utterly insensible in experiment. In like manner the 
directions of the planes of polarization according to the two theories, though 
not rigorously, are extremely nearly the same, the plane of polarization of a 
wave in which the vibrations are nearly transyersal being defined as that 
containing the direction of propagation and the direction of vibration, in har- 
mony with the previously established definition for the case of strictly trans- 
versal vibrations. 
Hence as far as regards the laws of double refraction of the two waves 
which alone are supposed to relate to the visible phenomenon, and of the 
accompanying polarization, this theory, by the aid of the forced relations (2), 
is very successful. I am not now discussing the generality, or, on the con- 
trary, the artificially restricted nature, of the fundamental suppositions as to 
the state of things, but only the degree to which the results are in accordance 
with observed facts. But as regards the third wave the case is very different. 
That theory should point to the necessary existence of such a waye consisting 
of strictly normal vibrations, and yet to which no known phenomenon can be 
referred, is bad enough; but in the present theory the vibrations are not 
even strictly normal, except for waves in a direction perpendicular to any one 
of the principal axes. In Iceland spar, for instance, for waves propagated in 
a direction inclined 45° to the axis, it follows from the numerical values of 
the refractive indices for the fixed line D given by Rudberg that the two 
vibrations in the principal plane which can be propagated independently of 
each other are inclined at angles of 9° 50’ and 80° 10’, or say 10° and 80°, to 
the wave-normal. We can hardly suppose that a mere change of inclination 
in the direction of vibration of from 10° to 80° with the wave front makes all 
the difference whether the wave belongs to a long-known and evident pheno- 
menon, no other than the ordinary refraction in Iceland spar, or not to any 
visible phenomenon at all. 
It is true that before there can be any question of the third wave’s being 
perceived it must be supposed excited, and the means of exciting it consist in 
the incident vibrations in air, which by hypothesis are strictly transversal. 
Hence we have to inquire whether the intensity of the third wave is such as 
to lead us to expect a sensible phenomenon answering to it. This leads us to 
the still more uncertain subject of the intensity of light reflected or refracted 
at the surface of a crystal—more uncertain because it not only depends on 
the laws of internal propagation, and inyolves all the hypotheses on which 
these laws are theoretically deduced, but requires fresh hypotheses as to the 
state of things at the confines of two media, introducing thereby fresh elements 
of uncertainty. But for our present purpose no exact calculation of intensities 
is required; a rough estimate of the intensity of the nearly normal vibrations 
is quite sufficient. 
In order to introduce as little as possible relating to the theory of the in- 
tensity of reflected and refracted light, suppose the incident light to fall per- 
pendicularly on the surface of a crystal, and let this be a surface of Iceland 
spar cut at an inclination of 45° to the axis. For a cleavage plane the result 
would be nearly the same. Let the incident light be polarized, and the 
vibrations be in the principal plane, which therefore, according to the theory 
* Annales de Chimie, tom. xlviii. p. 254 (1831). 
