104 Mr. MacCullagh on the Laws of Reflexion 
rallel to either semiaxis of this ellipse, the index of refraction 
for the wave being equal to the other semiaxis. ‘This is 
Fresnel’s law of double refraction; and the theory which led 
him to it makes it necessary to admit that the vibrations of 
the wave are perpendicular to its plane of polarization; 
whereas, according to the views which I have adopted, the 
vibrations of the wave are parallel to its plane of polarization, 
and to one semiaxis of the elliptic section, while its index of 
refraction is equal to the other semiaxis. ‘These views nearly 
agree with the theory of M. Cauchy, according to whom 
the vibrations of polarized light are parallel to its plane of 
polarization, but inclined at small angles to the plane of the 
wave in crystallized media, instead of being exactly parallel 
to the latter plane, as I have supposed them to be. Besides, 
the theory of M. Cauchy, founded on the six equations of 
pressure in a crystallized medium, implies the existence of a 
third ray of feeble intensity, ‘and for the other two rays gives 
a law somewhat different:from that of Fresnel. Being obliged, 
in order to account for your experiments, to abandon the 
physical ideas of Fresnel, and to approximate towards those 
of M. Cauchy, I was embarrassed by this third ray; and wish- 
ing to get rid of it, as well as of the slight deviations from 
the symmetrical law of Fresnel, 1 adopted the expedient of 
altering the equations of pressure, in such a way as to make 
them afford only two rays, and give a law of refraction exactly 
the same as Fresnel’s. The equations which I found to an- 
swer this purpose are the following: 
belt Ze Me ee 
Bre 0 ned aes 
GE e(esi se st) Vg 
Di ar i +a) ve 
Se 524 5*) V9 
F = c* ( A SLR 
In these equations, the axes of coordinates are perpendicu- 
lar to each other, and parallel to the three principal directions 
