Mechanical Theory of Circular and Elliptic Polarization. 407 
The foregoing investigation does not differ materially from that 
which I had recourse to in the beginning of the year 1836. To 
render the proof more easily intelligible, and to get rid of M. Cauchy’s 
‘third ray,” which has no existence in the nature of things, I have 
suppressed the normal vibrations; a procedure which is not, in ge- 
neral, allowable on the principles of M. Cauchy. It will readily 
appear, however, that this simplification still leaves the demonstra- 
tion perfectly rigorous in the case of circular vibrations, and does 
not affect its force when the vibrations are elliptical. For in the 
rotatory fluids it is obvious that the normal vibrations, supposing such 
to exist, must, by reason of the symmetry which the fluid constitu- 
tion requires, be independent of the transversal vibrations, and sepa- 
rable from them, so that the one kind of vibrations may be supposed 
to vanish when we wish merely to determine the laws of the other. 
The equations (2.) are, therefore, quite exact in this case; and they 
are also exact in the case of a ray passing along the axis of quartz, 
since such a ray is not experimentally distinguishable from one trans- 
mitted by a rotatory fluid, and its vibrations must consequently be 
subject to the same kind of symmetry. In these two cases, there- 
fore, it is rigorously proved that the values of k, which ought to be 
equal to plus and minus unity, are imaginary, and equal to + “—1. 
And if we now take the most general case with regard to quartz, 
and suppose that the ray, which was at first coincident with the axis 
of the crystal, becomes gradually inclined to it, the values of k must 
evidently continue to be imaginary, until such an inclination has 
been attained that the two roots of equation (5.) become possible 
and equal, in consequence of the increased magnitude of the coefii- 
cient of the second term. Supposing the last term of that equation 
to remain unchanged, this would take place when the coefficient of 
k (without regarding its sign) became equal to the number 2, and 
the values of & each equal to unity, both values being positive or 
both negative. The vibrations which before were impossible, would, 
at this inclination, suddenly become possible; they would be cir- 
cular, which is the exclusive property of vibrations transmitted along 
the axis ; and they would have the same direction in both rays, which 
is not a property of any vibrations that are known to exist. At 
greater inclinations the vibrations would be elliptical, but they would 
still have the same direction in the two rays. These results would 
not be sensibly altered by regarding the equation (5.) as only ap- 
proximate in the case of rays inclined to the axis; for the last term 
of that equation, if it does not remain the same, can never differ 
much from unity; since it must become exactly equal to unity, 
whatever be the direction of the ray, when the crystalline structure 
is supposed to disappear, and the medium to become a rotatory 
fluid. 
That a theory involving so many inconsistencies should have been 
advanced by a person of M. Cauchy’s reputation, would, perhaps, 
appear very extraordinary, if we did not recollect that it was un- 
avoidably suggested by the general principles which he had pre- 
viously adopted, and which were supposed, not merely by himself, 
