234 REPORT—1885. 
are left invariable, so that the ether equations remain unaltered, and the 
matter equations become— 
a?U 
dU 
Yae 
d 
= ta) 4 - ‘ ‘ . (58) 
and similar equations with a? and a3”. It has been shown by him that 
for a transparent medium the velocity is given by 1/r, where r is a 
radius, drawn in the direction of displacement of the surface— 
2 24 2 a? y? ! 2 = 9 2 hae = 
(+y 48-1) (sete ity i)=* +y+2 (59) 
1 a Saas 
and the directions of vibration are the axes of a section of this surface by 
the wave. 
These results are at variance with experiment, which requires that the 
wave surface should be that of Fresnel, and no reason is assigned in 
the paper for making a” rather than (3? or y? a function of the direction. 
Circular polarisation and the rotation of the plane of polarisation ! 
are also treated of by introducing into the equation for U the term 
— 20 cos os and into the equation for V, 20 cos ies where 6 de- 
pends on the strength of the magnetic force, and a is the angle 
between its direction and the axis of z. 
From this it follows that the rotation is proportional to 
a b 
tpt ; 
and the results of calculation agree fairly well with Verdet’s experiments.? 
For the rotation of sugar terms of the same kind, but without the 
cos a, are introduced. 
It has been shown long since, by Airy,? Neumann, and MacCullagh, 
that such terms in the equations would lead to results in fair agreement 
with experiment, and Lommel does not attempt any other justification of 
their existence than that the results they lead to are in agreement with 
experiment. Similar remarks apply to his paper on the properties of 
quartz,* in which the same terms are added to the differential equations 
already found for a crystalline media. The two waves travelling in any 
given direction inclined at an angle 6 to the axis are elliptically polarised. 
The elliptic paths of the particles are similar ; their ratio is given by— 
dy cos? 8 
Sr ary es 9 eee Wile 60 
Yb sin? 0 + {b? sin! 0 + do? cos! 6}! (60) 
and the difference of phase between the two by 
d? = b? sin‘ 0 + dy? cost @ . . a i 
where b and dy are functions of the refractive indices and wave lengths. 
The axial rotation is given by— 
N?—1)? 
Be ath 
? Lommel, ‘Theorie der Dehnung der Polarisationsebene,’ Wied. Ann. t. xiv. p. 523. 
2 Verdet, Ann. de Chim. (3), t. 69, p. 471. 
3 Airy, Phil. Mag. June, 1846; Neumann, Die magnetischen Dehnungen, Halle, 
1863 ; McCullagh, Roy. Irish Trans. 
* Lommel, ‘ Theorie der elliptischen Doppelbrechung,’ Wied. Ann. t. xv. p. 378. 
