MacCullagh and Neumann

Cauchy’s theories resemble Fresnel’s in postulating types of elastic solid which do not exist, and for whose assumed properties no dynamical justification is offered.

The same objection applies, though in a less degree, to the original form of a theory of reflexion and refraction which was discovered about this time[24] almost simultaneously by James MacCullagh (b. 1809, d. 1847), of Trinity College, Dublin, and Franz Neumann (b. 1799, d. 1895), of Königsberg.

To these authors is due the merit of having extended the laws of reflexion to crystalline media.

But the principles of the theory were originally derived in connexion with the simpler case of isotropic media, to which our attention will for the present be confined.

MacCullagh and Neumann felt that the great objection to Fresnel’s theory of reflexion was its failure to provide for the continuity of the normal component of displacement at the interface between two media.

A discontinuity in this component could not exist in any true elastic-solid theory, since it would imply that the two media do not remain in contact.

Accordingly, they made it a fundamental condition that all 3 components of the displacement must be continuous at the interface. They found that the sine-law and tangent-law can be reconciled with this condition only by supposing that the aether-vibrations are parallel to the plane of polarization: which supposition they accordingly adopted.

In place of the remaining 3 true boundary-conditions, however, they used only a single equation. It was derived by assuming:

• that transverse incident waves give rise only to transverse reflected and refracted waves, and
• that the conservation of energy holds for these—i.e. that the masses of aether put in motion, multiplied by the squares of the amplitudes of vibration, are the same before and after incidence.

This is the same device as had been used previously by Fresnel. But the principle is unsound as applied to an ordinary elastic solid; for in such a body the refracted and reflected energy would in part be carried away by longitudinal waves.

In order to obtain the sine and tangent laws, MacCullagh and Neumann found it necessary to assume that the inertia of the luminiferous medium is everywhere the same, and that the differences in behaviour of this medium in different substances are due to differences in its elasticity. The two laws may then be deduced in much the same way as in the previous investigations of Fresnel and Cauchy.

Although to insist on continuity of displacement at the interface was a decided advance, the theory of MacCullagh and Neumann scarcely showed as yet much superiority over the quasi-mechanical theories of their predecessors.

MacCullagh himself expressly disavowed any claim to regard his theory, in the form to which it had then been brought, as a final explanation of the properties of light. “If we are asked,” he wrote, “what reasons can be assigned for the hypotheses on which the preceding theory is founded, we are far from being able to give a satisfactory answer. We are obliged to confess that, with the exception of the law of vis viva, the hypotheses are nothing more than fortunate conjectures.

These conjectures are very probably right, since they have led to elegant laws which are fully borne out by experiments; but this is all we can assert respecting them.

We cannot attempt to deduce them from first principles; because, in the theory of light, such principles are still to be sought for. It is certain, indeed, that light is produced by undulations, propagated, with transversal vibrations, through a highly elastic aether; but the constitution of this aether, and the laws of its connexion (if it has any connexion) with the particles of bodies, are utterly unknown.”

The needful reformation of the elastic-solid theory of reflexion was effected by Green, in a paper[25] read to the Cambridge Philosophical Society in December, 1837.

Green was inferior to Cauchy as an analyst. But he was superior in physical insight.

Instead of designing boundary-equations for the express purpose of yielding Fresnel’s sine and tangent formulae, he determined the conditions which are actually satisfied at the interfaces of real elastic solids.

These he obtained through general dynamical principles.

In an isotropic medium which is strained, the potential energy per unit volume due to the state of stress is:

(where ρ denotes the density), the equation of motion may be deduced.

But this method does more than merely furnish the equation of motion

or,

which had already been obtained by Cauchy; for it also yields the boundary-conditions which must be satisfied at the interface between two elastic media in contact; these are, as might be guessed by physical intuition, that the three components of the displacement[26] and the three components of stress across the interface are to be equal in the two media.

If the axis of x be taken normal to the interface, the latter 3 quantities are