ON OPTICAL THEORIES. 165 
already given by Green.! And in cases in which the axes can only be 
turned together about the origin, a third coefficient comes in, in the form 
of terms, such as 
dw dv 
Caer ): 
In 1849? Cauchy propounded the idea that the ether atoms in a body 
such as a crystal are disposed, as it were, in shells round the matter atoms 
in such a manner as to have different elastic properties at different points 
of the same shell; the shells, however, are regularly placed, and the 
properties of the ether repeat themselves at similar points in the different 
shells. It results from this that the constants in the equations of motion 
will be periodic functions of the equilibrium positions of the molecules, 
and for optical effects we have to do with the average displacement over 
a small volume of the medium.* 
The general equations established by Cauchy lead to a normal wave 
travelling with a velocity equal to / A/p. According to his earlier theory, 
resting on the law of action between the molecules of ether, A and B are 
not independent, and it is possible by suitably choosing the law of force 
to make A vanish or even be negative. The theory‘ of reflexion and 
refraction led him to conclude that A was a small negative quantity, so 
that the normal disturbance ceases to be propagated as such. 
§ 4. Cauchy’s work was continued by Briot,’ starting from the 
equations of motion deduced from the mutual action between two par- 
ticles of ether, and the supposition suggested by Cauchy that the ether 
within a crystal is in a state of unequal strain. In treating of dispersion 
Briot points out that it cannot be explained in the manner originally 
suggested by Cauchy, for there is no reason why the terms in his differ- 
ential equations from which it arises should be insensible ina vacuum if they 
are sensible in ordinary transparent media. He therefore makes it depend 
on terms arising from a periodic distribution of the ether within material 
bodies, and shows that to obtain Cauchy’s dispersion formula the law of 
action between the molecules must be as the inverse sixth power of the 
distance. In his memoir on reflexion and refraction, however, he adopts 
Cauchy’s views as to the disappearance of the normal wave, and this is 
quite inconsistent with the above law, while the ether and matter mole- 
cules must attract each other with a force varying as the inverse square 
of the distance. 
§ 5. The problem of reflexion and refraction for both isotropic and 
crystalline bodies is treated of in a memoir published in 1866-67,° start- 
ing from Cauchy’s principle of continuity, to which he gives an extended 
meaning in the second memoir. He at first supposes the vibrations in 
the crystal to be rigorously in the plane of the wave, and, adopting 
MacCullagh’s methods of the uniradial direction, arrives at his equations. 
The work is then extended to the general case in which the vibrations 
1 See p. 161. 
* C. R. t. xxix. pp. 641, 644, 728, 762; t. xxx. p. 27. 
* For the further development of this by M. Sarrau, see p. 174. 
* C. R. t. ix. pp. 677, 727, 765. On this point cf. Green’s theory. See also 
Stokes’s, Brit. Assoc. Report, 1862, and pp. 170-195. 
* Briot, Hssais sur la théorie mathématique de la lumiére. Paris, 1864. 
* Liouville’s Journal, t. xi. p. 305; t. xii. p. 185. 
