Cauchy's theories

A further objection to Cauchy’s theories is that the relations between the constants do not have any simple physical interpretation. They merely forced the formulae to conform with the results of experiment.

Further difficulties will appear when we compare the properties of the aether in crystal-optics with those which account for reflection and refraction.

Any work on refraction needs to assign a cause for the existence of refractive indices, i.e. for the variation in the speed of light from one body to another.

Huygens suggested that transparent bodies consist of hard particles which interact with the aethereal matter, modifying its elasticity.

Cauchy followed this, assuming that:

• the density ρ of the aether is the same in all media
• its rigidity n varies from one medium to another.

Let:

• the axis of x be taken at right angles to the surface of separation of the media
• the axis of z be parallel to the intersection of this interface with the incident wave-front

Suppose that the incident vibration is executed at right angles to the plane of incidence, so that it may be represented by:

where i denotes the angle of incidence.

The reflected wave may be represented by:

and the refracted wave by:

where r denotes the angle of refraction, and n′ the rigidity of the second medium.

To obtain the conditions satisfied at the reflecting surface, Cauchy assumed (without assigning reasons) that the x- and y-components of the stress across the xy-plane are equal in the media on either side the interface. This implies in the present case that the quantities

are to be continuous across the interface. So we have

Eliminating f′1, we have

This is Fresnel’s sine-law for the ratio of the intensity of the reflected ray to that of the incident ray.

The light to which it applies is that which is polarized parallel to the plane of incidence.

Thus, Cauchy was driven to conclude that, in order to satisfy the known facts of reflexion and refraction, the vibrations of the aether must be supposed executed at right angles the plane of polarization of the light.

He discussed the case of a vibration performed in the plane of incidence in the same way.

It was found that Fresnel’s tangent-law could be obtained by assuming that ex and the normal pressure across the interface have equal values in the two contiguous media.

The theory thus had many difficulties.

1. The identification of the plane of polarization with the plane at right angles to the direction of vibration was contrary to the only theory of crystal-optics which Cauchy had as yet published.
2. No reasons were given for the choice of the conditions at the interface.

Cauchy’s motive in selecting these conditions was to fulfill Fresnel’s sine-law and tangent-law. But the results are inconsistent with the true boundary-conditions, which were given later by Green.

It is probable that the results of the theory of reflexion had much to do with Cauchy’s decision [20] to reject the first theory of crystal-optics in favour of the second.

After 1836, he consistently adhered to the view that the vibrations of the aether are performed at right angles to the plane of polarization.

In that year, he tried again to frame a theory of reflexion,[21] based on that assumption, and on the following boundary-conditions:

• at the interface between 2 media curl e is to be continuous, and
• (taking the axis of x normal to the interface)

is also to be continuous.

Again, there were no satisfactory reasons for the choice of the boundary-conditions. The continuity of e itself across the interface is not included amongst the conditions chosen, they are obviously open to criticism.

But they lead to Fresnel’s sine- and tangent-equations, which correctly express the actual behaviour of light.[22]

Cauchy remarks that in order to justify them it is necessary to abandon the assumption of his earlier theory, that the density of the aether is the same in all material bodies.

Neither in this nor in Cauchy’s earlier theory of reflexion is any trouble caused by the appearance of longitudinal waves when a transverse wave is reflected, for the simple reason that he assumes the boundary-conditions to be only four in number; and these can all be satisfied without the necessity for introducing any but transverse vibrations.

These features bring out the weakness of Cauchy’s method of attacking the problem.

His object was to derive the properties of light from a theory of the vibrations of elastic solids. At the outset he had already in his possession the differential equations of motion of the solid, which were to be his starting-point, and the equations of Fresnel, which were to be his goal.

It only remained to supply the boundary-conditions at an interface, which are required in the discussion of reflexion, and the relations between the elastic constants of the solid, which are required in the optics of crystals. Cauchy seems to have considered the question from the purely analytical point of view.

Given certain differential equations, what supplementary conditions must be adjoined to them in order to produce a given analytical result?

The problem when stated in this form admits of more than one solution, and hence it is not surprising that within the space of 10 years the great French mathematician produced two distinct theories of crystal-optics and three distinct theories of reflexion,[23] almost all yielding correct or nearly correct final formulae, and yet mostly irreconcilable with each other, and involving incorrect boundary-conditions and improbable relations between elastic constants.