IN OPTICAL THEORIES. 237 
Where A, =A, + as A, and A,’ being functions of a, b, c, ete., and P 
is a linear function of p and the differential coefficients oe ete. 
d 
The equations in this form may be compared with Green’s, which differ 
from them only in the facts that his coefficients of d?w/dt?, d?y/dt?, and 
@z/di? are the same, and his other coefficients are independent of the 
time. Voigt’s equations, in fact, include both Green’s and those given by 
Lord Rayleigh. 
Let r be the period of vibration, and denote m, — 7,77 by T,, ete. ; then 
it is shown that if we assume the relation - + gis + tes 0, in order to 
ye 
dy dz 
obtain Fresnel’s wave surface at all the condition T,; =T,=T,;=T is 
necessary. 
These equations being satisfied, the other relations required to give 
Fresnel’s construction on either assumption as to the connection between 
the plane of polarisation and the direction of vibration are those given by 
Green, with the addition that since in Voigt’s coefficients the period is 
involved, and since Fresnel’s construction holds for all wave lengths, each 
of Green’s relations splits into two. 
A difficulty as to the meaning of the constants leads Voigt to prefer 
Neumann and MacCullagh’s theory as to the position of the plane of 
polarisation. To obtain Fresnel’s original construction it is necessary to 
suppose B,, to be different from B,,, and this would imply that elastic 
reactions are brought into play by rotating an element of ether as a whole 
without dilatation ; that, in the ordinary notation of elastic solids, T,, is 
different from T.,. If we treat this as out of the question, then B,, must 
be equal to B,,, and Fresnel’s original construction for the plane of polari- 
‘sation is impossible. 
Circular polarisation is explained by the terms introduced by /,, f;, fs, 
and f, of above,' but the terms to which /, and f, would give rise are 
omitted as not necessary to explain any known phenomena, and the 
equations in an isotropic medium become— 
Ob ay Ob dv p' dv 
Tf Ee ie eae 
ete. ; the rotation produced by a thickness ¢ of the medium will be— 
2 
(m+ cee (e+ a) 
> 
eee apes se nec ae 
The same terms are then applied to a crystal, and the case of a uniaxial 
erystal such as quartz is worked out in full. 
The equation to determine the velocity in a direction making an angle 
6 with the axis is found to be— 
12 
(u8— a2) (w? — a? cost 0 — D4 sin? 0) =1( pu? 2 ) G3 
a and 6 being the velocities at right angles to the axis. 
_ This paper then gives a consistent account of the propagation of light 
in all known transparent bodies. We proceed to deal with the problem 
1 See p. 231. 
