crystalline Reflexion and Refraction. 27 
and are commonly known by the name of axes of elasticity. Thus the existence 
of these axes is proved without any hypothesis respecting the arrangement of the 
particles of the medium. The constants a, 6, c are the three principal velocities 
of propagation, as we shall see in the next section. 
Having arrived at the value of V, we may now take it for the starting point 
of our theory, and dismiss the assumptions by which we were conducted to it. 
Supposing therefore, in the first place, that a plane wave passes through a crystal, 
we shall seek the laws of its motion from equations (1) and (2), which contain 
everything that is necessary for the solution of the problem. The laws of 
propagation, as they are called, will in this way be deduced, and they will be 
found to agree exactly, so far as magnitudes are concerned, with those discovered 
by Fresnel; but the direction of the vibrations in a polarised ray will be diffe- 
rent from that assigned by him. In the second place, we shall investigate the con- 
ditions which are fulfilled when light passes out of one medium into another, and 
we shall thus obtain the laws of reflexion and refraction at the surface of a crystal. 
SECT. IV.—PROPAGATION OF LIGHT IN A CRYSTALLIZED MEDIUM.—LAWS OF 
FRESNEL.—ALTERATION REQUIRED TO BE MADE IN THEM.—WAVE-SURFACE, 
INDEX-SURFACE, AND THEIR PROPERTIES. 
The principal axes of the crystal being the axes of a, y, z, we have, by equa- 
tion (2), 
— 6V = a’xéx 4 Byéy + c’zéz, (3) 
or, by taking the variations from formule (c), and interchanging the characte- 
ristics d and 6, 
» (din doe » (dee dé » (dc=e don 
—8V = ax(—!_ 2s pg | (pesto aa ee) BA ey le eocweet 
ax(F ij) ae ae) +eu(F ds) 
and if we substitute this value in equation (1), and then integrate by parts the 
right-hand member, in order to get rid of the differential coefficients of the 
variations, we shall obtain 
RE 2 & a A 
SS dadydz (] 6& + = oy + aS a) — 
Sdydz (c’ zn — b* vex) +S dads (a? xé¢ — c° 28) + YS dady (b° vee — a? xéy) (4) 
Z ody 
+ SB dedyde } (°F — po bé+ (aS _— oe) oy (0 aaa) ee t. 
E2 
