138 Professor Hamitton’s Third Supplement 
ray by this ordinary law, if we take as the refracting index of the crystal the 
reciprocal 5>' of the mean semiaxis of elasticity. It is evident hence that if the inter- 
nal cone emerge at a new plane face, it will emerge a cylinder, whether the two faces 
be parallel or inclined, that is, whether the crystal be a plate or a prism. 
Theory of Conical Polarisation. Lines of Vibration. These Lines, on Fres- 
NEL’s /Vave, are the Intersections of Two Series of Concentric and Co-axal 
Ellipsoids. 
30. A given direction of a wave-normal in a biaxal crystal corresponds in general 
to two directions of vibration, and therefore to two planes of polarisation, determined 
by the equations (#""), namely one for each of the two values «,’, ,” of the square 
of the normal velocity deduced by (G") from the given system of ratios of o, 7, v; 
and these two directions of vibration, or the two planes of polarisation, that is, the 
two normal planes of the wave perpendicular to these vibrations, are perpendicular to 
each other, since we can easily deduce from (G™) the following relation between 
wr) W's 
o® E v? 
7 
rerio + Wwe + Gaa@ee =9? PMY 
which general rectangularity of the two vibrations on any one plane wave has been 
otherwise established by Fresnet, and is an important result of his theory. But | 
besides this general double polarisation connected with the general double refraction 
in biaxal crystals, we may consider two other kinds which may be called conical polar- 
isation, connected with the two kinds of conical refraction, which were pointed out in 
the foregoing number. 
To examine the law of the conical polarisation connected with the internal conical 
refraction, and therefore with the planes of circular contact, we may employ the 
co-ordinates 7, y,, 2,, defined by (AX), and thus transform the general equations of 
polarisation (4°) (#"") into the following equally general, 
, 7 Q 
W. OL, +0 a oz, 
aT tN OY, — 0,02, +0’, OF 
7 ; *—a@)= =4 (W* —8) = —* 4 | (w*-—¢ 
WF, +W GV, & ) By (w ) — Wao, +0, (w ce), (E*) 
o 82, oF 7,Y, + v8z,,= 0) 5 
which give, for the projection of a vibration on the plane x, y, of single normal 
velocity, the rigorous formula 
oY, ae (w* —a’) (w? —c?) oi, 
oar” Lesa v, Vae—B Vc to, (w?t+l?—at—ct) ” 
(F*) 
