On the Laws of the Double Refraction of Quartz. By James MacCutvacu, 
Fellow of Trinity College, Dublin. 
Read February 22, 1836. 
The singular laws of the double refraction of quartz, which have been discovered 
by the successive researches of Arago, Biot, Fresnel, and Airy, are known merely as 
so many independent facts ; they have not been connected by a theory of any kind. 
I propose, therefore, to show how these laws may be explained hypothetically, by 
introducing differential coefficients of the third order into the equations of vibratory 
motion. 
Suppose a plane wave of light to be propagated within a crystal of quartz. Let the 
coordinates x, y, z, of a vibrating molecule be rectangular, and take the axis of z per- 
pendicular to the plane of the wave, and the axis of y perpendicular to the axis of the 
crystal. Let us admit that the vibrations are accurately in the plane of the wave, and 
of course parallel to the plane of wy. Then, using § and 7 to denote, at any time ¢, 
the displacements parallel to the axes of x and y respectively, we shall assume the two 
following equations for explaining the laws of quartz :— 
a ae a 
ene Soden a1) 
dn <_ d°n BE f 
ee ae ame (2) 
The peculiar properties of this crystal depend on the constant C. When C=0, the 
third differentials disappear, and the equations are reduced to the ordinary form, in 
which state they ought to agree with the common equations for uniaxal crystals. 
Hence, putting a for the reciprocal of the ordinary index, b for the reciprocal of the 
extraordinary, and ¢ for the angle made by the axis of z with the axis of the crystal, 
we must have 
A=a, « B=a’—(a’—2’) sin’ ¢, - (3.) 
supposing the velocity of propagation in air to be unity. 
