Lord Kelvin's Theory

It is difficult to picture how the molecules of ponderable matter act on the aether so as to produce the initial strain required by this theory.

Lord Kelvin[47] suggested that the aether may pervade the atoms of matter so as to occupy space jointly with them. Its interaction with them may consist in attractions and repulsions exercised throughout the regions interior to the atoms.

These forces may be so large compared with those called into play in free aether that the resistance to compression may be overcome. The aether may be (say) condensed in the central region of an isolated atom, and rarefied in its outer parts.

A crystal may be supposed to consist of a group of spherical atoms in which neighbouring spheres overlap each other; in the central regions of the spheres the aether will be condensed, and within the lens-shaped regions of overlapping it will be still more rarefied than in the outer parts of a solitary atom, while in the interstices between the atoms its density will be unaffected.

In consequence of these rarefactions and condensations, the reaction of the aether on the atoms tends to draw inwards the outermost atoms of the group, which, however, will be maintained in position by repulsions between the atoms themselves, and thus we can account for the pull which, according to the present hypothesis, is exerted on the aether by the ponderable molecules of crystals.

Analysis similar to that of Cauchy’s and Green’s Second Theory of crystal-optics may be applied to explain the doubly refracting property which is possessed by strained glass; but in this case the formulae derived are found to conflict with the results of experiment.

The discordance led Kelvin to doubt the truth of the whole theory. “After earnest and hopeful consideration of the stress theory of double refraction during fourteen years,” he said,[48] “I am unable to see how it can give the true explanation either of the double refraction of natural crystals, or of double refraction induced in isotropic solids by the application of unequal pressures in different directions.”

It is impossible to avoid noticing throughout all Kelvin’s work evidences of the deep impression which was made upon him by the writings of Green. The same may be said of Kelvin’s friend and contemporary Stokes.

It is no exaggeration to describe Green as the real founder of that “Cambridge school” of natural philosophers, of which Kelvin, Stokes, Lord Rayleigh, and Clerk Maxwell were the most illustrious members in the latter half of the nineteenth century, and which is now led by Sir Joseph Thomson and Sir Joseph Larmor. In order to understand the peculiar position occupied by Green, it is necessary to recall something of the history of mathematical studies at Cambridge.

The century which elapsed between the death of Newton and the scientific activity of Green was the darkest in the history of the University. It is true that Cavendish and Young were educated at Cambridge.

But they, after taking undergraduate courses, removed to London, In the entire period the only natural philosopher of distinction who lived and taught at Cambridge was Michell; and for some reason which at this distance of time it is difficult to understand fully, Michell’s researches seem to have attracted little or no attention among his collegiate contemporaries and successors, who silently acquiesced when his discoveries were attributed to others, and allowed his name to perish entirely from Cambridge tradition.

A few years before Green published his first paper, a notable revival of mathematical learning swept over the University; the fluxional symbolism, which since the time of Newton had isolated Cambridge from the continental schools, was abandoned in favour of the differential notation, and the works of the great French analysts were introduced and eagerly read.

Green undoubtedly received his own early inspiration from this source, but in clearness of physical insight and conciseness of exposition he far excelled his masters; and the slight volume of his collected papers has to this day a charm which is wanting to the voluminous writings of Cauchy and Poisson. It was natural that such an example should powerfully influence tho youthful intellects of Stokes—who was an undergraduate when Green read his memoir on double refraction to the Cambridge Philosophical Society—and of William Thomson (Kelvin), who came into residence two years afterwards.[49]

In spite of the advances which were made in the great memoirs of the year 1839, the fundamental question as to whether the aether-particles vibrate parallel or at right angles to the plane of polarization was still unanswered.

More light was thrown on this problem ten years later by Stokes’s investigation of Diffraction.[50] Stokes showed that on almost any conceivable hypothesis regarding the aether, a disturbance in which the vibrations are executed at right angles to the plane of diffraction must be transmitted round the edge of an opaque body with less diminution of intensity than a disturbance whose vibrations are executed parallel to that plane.

It follows that: when light, of which the vibrations are oblique to the plane of diffraction, is so transmitted, the plane of vibration will be more nearly at right angles to the plane of diffraction in the diffracted than in the incident light. Stokes himself performed experiments to test the matter, using a grating in order to obtain strong light diffracted at a large angle, and found that when the plane of polarization of the incident light was oblique to the plane of diffraction, the plane of polarization of the diffracted light was more nearly parallel to the plane of diffraction. This result, which was afterwards confirmed by L. Lorenz,[51] appeared to confirm decisively the hypothesis of Fresnel, that the vibrations of the aethereal particles are executed at right angles to the plane of polarization.

Three years afterwards Stokes indicated[52] a second line of proof leading to the same conclusion. It had long been known that the blue light of the sky, which is due to the scattering of the sun’s direct rays by small particles or molecules in the atmosphere, is partly polarized. The polarization is most marked when the light comes from a part of the sky distant 90° from the sun, in which case it must have been scattered in a direction perpendicular to that of the direct sunlight incident on the small particles; and the polarization is in the plane through the sun.

If, then, the axis of y be taken parallel to the light incident on a small particle at the origin, and the scattered light be observed along the axis of x, this scattered light is found to be polarized in the plane xy. Considering the matter from the dynamical point of view, we may suppose the material particle to possess so much inertia (compared to the aether) that it is practically at rest.

Its motion relative to the aether, which is the cause of the disturbance it creates in the aether, will therefore be in the same line as the incident aethereal vibration, but in the opposite direction. The disturbance must be transversal, and must therefore be zero in a polar direction and a maximum in an equatorial direction, its amplitude being, in fact, proportional to the sine of the polar distance.

The polar line must, by considerations of symmetry, be the line of the incident vibration. Thus we see that none of the light scattered in the x-direction can come from that constituent of the incident. light which vibrates parallel to the x-axis, so the light observed in this direction must consist of vibrations parallel to the z-axis. But we have seen that the plane of polarization of the scattered light is the plane of xy; and therefore the vibration is at right angles to the plane of polarization.[53]

The phenomena of diffraction and of polarization by scattering thus agreed in confirming the result arrived at in Fresnel’s and Green’s theory of reflexion. The chief difficulty in accepting it arose in connexion with the optics of crystals.

As we have seen, Green and Cauchy were unable to reconcile the hypothesis of aethereal vibrations at right angles to the plane of polarization with the correct formulae of crystal-optics, at any rate so long as the aether within crystals was supposed to be free from initial stress. The underlying reason for this can be readily In a crystal, where the elasticity is different in different. directions, the resistance to distortion depends solely on the orientation of the plane of distortion, which in the case of light. is the plane through the directions of propagation and vibration.

Now it is known that for light propagated parallel to one of the axes of elasticity of a crystal, the velocity of propagation depends only on the plane of polarization of the light, being the same whichever of the two axes lying in that plane is the direction of propagation. Comparing these results, we see that. the plane of polarization must be the plane of distortion, and therefore the vibrations of the aether-particles must be executed parallel to the plane of polarization.[54]

A way of escape from this conclusion suggested itself to Stokes,[55] and later to Rankine[56] and Lord Rayleigh.[57] What if the aether in a crystal, instead of having its elasticity different in different directions, were to have its rigidity invariable and its inertia different in different directions?

This would bring the theory of crystal-optics into complete agreement with Fresnel’s and Green’s theory of reflexion, in which the optical differences between media are attributed to differences of inertia of the aether contained within them.

The only difficulty lies in conceiving how aelotropy of inertia can exist; and all three writers overcame this obstacle by pointing out that a solid which is immersed in a fluid may have its effective inertia different in different directions. For instance, a coin immersed in water moves much more readily in its own plane than in the direction at right angles to this.

Suppose then that twice the kinetic energy per unit volume of the aether within a crystal is represented by the expression

and that the potential energy per unit volume has the same value as in space void of ordinary matter. The aether is assumed to be incompressible, so that div e is zero: the potential energy per unit volume is therefore

where p denotes an undetermined function of (x, y, z): the term in p being introduced on account of the kinematical constraint expressed by the equation

The equations of motion which result from this variational equation are

and two similar equations. It is evident that p resembles a hydrostatic pressure.

Substituting in these equations the analytical expression for a plane wave, we readily find that the velocity V of the wave is connected with the direction-cosines (λ, μ, ν) of its normal by the equation

When this is compared with Fresnel’s relation between the velocity and direction of a wave, it is seen that the new formula differs from his only in having the reciprocal of the velocity in place of the velocity.