Royal Society. 155 
their passage through a double-refracting crystal ; and thus that the 
forms of the rings, or isochromatic curves, which had previously been 
deduced only by approximate methods, may be determined generally. 
The next paper of Professor MacCullagh is that “ On the Double 
Refraction of Quartz* ;’ a subject which had engaged the attention, 
successively, of Biot, Fresnel, and Airy. ‘The first of these writers 
had determined experimentally the laws of rotatory polarization, 
which take place when a ray is transmitted along the axis of rock- 
crystal; and the second had shown that these laws were explained 
by the interference of two circularly polarized rays, which are trans- 
mitted along the axis with different velocities. The next step in this 
curious subject was made by Mr. Airy, who examined the peculiar 
phenomena of refraction by quartz in other directions, and showed 
that they were accounted for by the supposition of two elliptically 
polarized rays, the ratio of the axes of these elliptical vibrations va- 
rying with the inclination of the rays to the axis of the crystal. 
Lastly, Professor MacCullagh has shown that both the circular po- 
larization of the rays in the axis, and the elliptical polarization of the 
rays inclined to it, may be explained by a certain assumed form of 
the differential equations of vibratory movement, which not only 
links together the two classes of phenomena, but also affords a ma- 
thematical expression for their laws. The general theory, to be al- 
luded to presently, has enabled him to explain the origin of these as- 
sumed forms of the differential equations. 
The theory of reflexion at the surfaces of unerystallized media 
had been given by Fresnel, although apparently on erroneous prin- 
ciples. The more complex case of reflexion at the surfaces of cry- 
stals was left by him to his successors; and the discovery was made 
independently, and nearly at the same time, by Professor MacCul- 
laght and M. Newmann of KGnigsberg. The discovery is not 
only important in itself, as bringing within the domain of the wave- 
theory a large class of hitherto unexplained phenomena, but per- 
haps still more on account of the physical principles upon which it 
is based, and the constitution of the luminiferous ether which it ren- 
ders probable. Thus, it is assumed in this theory, in opposition to 
the hypothesis of Fresnel, that the vibrations are parallel to the plane 
of polarization, and that the density of the ether is the same in all 
media. These, together with the law of the vis viva, and the beauti- 
ful principle of the equivalence of vibrations (but half perceived by 
Fresnel), form the foundation of the theory of crystalline reflexion, 
and derive the highest probability from its accordance with pheno- 
mena. The results of the theory are embodied in geometrical con- 
structions of great elegance, which determine generally the plane of 
polarization of the reflected ray, and the amplitudes of the retlected 
and refracted vibrations. 
Hitherto the laws of reflexion at the separating surface of two me- 
* Transactions of the Royal Irish Academy, vol. xvii. 
+ ‘‘On the laws of crystalline reflexion and refraction.’”’ Transactions 
of the Royal Irish Academy, vol. xviii. This memoir has been honoured 
by the Medal of the Royal Irish Academy. [Phil. Mag. S,3.vol. xi, p. 134.] 
