166 REPORT— 1885. 
| 
are quasi-transversal, and it is shown how the simpler forms of the — 
equations are modified by this. ; 
Thus, for the uniradial directions in the case in which the longitudinal 
disturbance is supposed to be strictly normal to the wave, if y is the 
angle between the ray and the wave normal, 0, 6’, and 6, the azimuths 
of the planes of polarisation, measured from the plane of incidence, of the 
incident reflected and refracted waves, @ and 9’ the angles of incidence 
and refraction, and m a quantity depending on the angle between the 
plane of the wave and the direction of vibration, then— 
i Ulta 0 0 an lt i! 
tan 6 = tan 0’ cos (¢ — 9’) + cos 0’ sin (9 + "| (14) 
sonal Cee eee f 
tan 0, = tan 0’ cos(¢ + ¢’) cos 0’ sin (9 — 9’) 
é 
These formule agree with those of MacCullagh if we put m = 1. 
Chapter IV.—Etuipric Potarisation. Comparison oF REsuLTs. 
§ 1. The peculiar phenomena presented by quartz had been explained 
by Airy in 1831! on the assumption that the two waves were elliptically 
polarised. In 1836% MacCullagh made a further advance, and showed 
how the addition of certain terms to the differential equations of motion 
would lead to the elliptic polarisation required by Airy’s theory. The 
equations assumed by MacCullagh, for the existence of which he does 
not attempt to assign a mechanical reason, were— 
i acu ote 
dt dz dz if 
Me _ pie _ ohn se 
de dz dz 
Where A = a”, B = a? — (a? — 3b?) sin? 6, 
a and b being constants, and @ the angle between the optic axis and 
the wave normal—the axis of z. The two waves resulting from these 
equations are shown to be elliptically polarised, while their velocity is 
given by the equation 
: An? C? 
(w? — A) (uw? —B) ==, 
(16) 
dX being the wave length. The rotation of the plane of polarisation 
produced by the passage of a plane polarised ray through a plate of 
crystal cut at right angles to the axis, and of unit thickness, is 277C /a*n?. 
MacCullagh shows that the results of this hypothesis as to the form 
of the equations agree fairly with Airy’s experiments, and that the 
agreement would be made somewhat more close by the hypothesis that C 
varies slightly with 0. 
1 Airy, ‘On the Nature of the Two Rays produced by the Double Refraction of 
Quartz,’ Camb. Phil. Soc. Trans. vol. iv. pp. 79, 198. 
? MacCullagh, ‘On the Laws of the Double Refraction of Quartz,’ Ivish Trans. 
vol. xvii. p. 461. . 
