256 REPORT— 1885. 
same as that of the regular or average displacement, but the relations 
between , n, ¢ and &’, n’, ¢’ change rapidly as we pass from point to point of 
the element. 
The velocity of wave propagation is found by equating the maximum 
potential and kinetic energies of the medium. It is shown that the 
equations lead to Fresnel’s construction in the case of a crystal if the 
solenoidal condition be assumed, while the relation between yu the refractive 
index and \ the wave length is given by 
Lyon H is 27,H’ 
pe Dark? Ne 
(20) 
H, k, and H’ being constants. The objection which Briot made to 
Cauchy’s theory of dispersion may be made to this. We should expect 
dispersion ina vacuum as well as in ordinary transparent media. 
The properties of circularly polarising media are discussed in a second 
paper,! in which 2’, 7’, ¢’ are treated as linear functions of ¢, y, ¢ and their 
differential co-efficients ; and in a third paper the fundamental equations are 
re-established in rather a more general form than that given by Maxwell. 
The generality is gained partly by dealing with the average values of 
the various quantities, and partly by supposing that the relation between 
the E.M.F. and displacement is given by 
(y= 410) Eh]: =) SS See 
¢and J being two arbitrary functions, and [ | indicating that the average 
value is taken. In the simple theory ¢ is a constant, and equal to 4a/K, 
and wW zero, and this will not give dispersion. 
There seems, however, to be no reason—as has been pointed out by 
Professor Fitzgerald—against applying to the oscillations of the electro- 
magnetic field the methods and reasoning developed in the third part of 
this report. Almost the whole of the work can be translated into the 
language of the electro-magnetic theory at once. Periodic electric dis- 
placement in the ether will produce periodic electric displacement in the 
matter, and the relations between the two will depend on the ratio of the 
period of the ether vibrations to the possible free periods of the electric 
oscillations in the matter molecules ; and it is not difficult to see how the 
action between the two might depend on the relative electrical displace- 
ments and their differential coefficients. 
§ 2. Maxwell? has given a theory of the magnetic rotation of the plane 
of polarisation on this theory. He assumes (1) that the effect of mag- 
netic force is to set up molecular vortices in the medium; (2) that the 
components of the magnetic force obey the same law as the components 
of the strength of a vortex in hydrodynamics; and (3) that there arises 
in the value for the kinetic energy of the medium aterm of the form 
2C(aw, + Bwy+ yw3), ©), 2, 3 being the components of the angular 
velocity, and a, 3, y of the magnetic force. 
For the case of waves travelling parallel to z the kinetic energy is 
shown to be 
: : : , we 3d? 
cl Yo Aiea a5 th a 67) + oy(a ae et eo) . ° (22) 
1 J. W. Gibbs, American Journal of Science, vol. xxiii. June, 1882. 
2 Maxwell, Electricity and Magnetism, vol. ii. p. 40. 
