# A History of the Theories of Aether and Electricity/Chapter 5

CHAPTER V.
THE AETHER AS AN ELASTIC SOLID.
When Young and Fresnel put forward the view that the vibrations of light are performed at right angles to its direction of propagation, they at the same time pointed out that this peculiarity might be explained by making a new hypothesis regarding the nature of the luminiferous medium; namely, that it possesses the power of resisting attempts to distort its shape. It is by the possession of such a power that solid bodies are distinguished from fluids, which offer no resistance to distortion; the idea of Young and Fresnel may therefore be expressed by the simple statement that the aether behaves as an elastic solid. After the death of Fresnel this conception was developed in a brilliant series of memoirs to which our attention must now be directed.
The elastic-solid theory meets with one obvious difficulty at the outset. If the aether has the qualities of a solid, how is it that the planets in their orbital motions are able to journey through it at immense speeds without encountering any perceptible resistance? This objection was first satisfactorily answered by Sir George Gabriel Stokes (b. 1819, d. 1903), who remarked that such substances as pitch and shoemaker's wax, though so rigid as to be capable of elastic vibration, are yet sufficiently plastic to permit other bodies to pass slowly through them. The aether, he suggested, may have this combination of qualities in an extreme degree, behaving like an elastic solid for vibrations so rapid as those of light, but yielding like a fluid to the much slower progressive motions of the planets.
Stokes's explanation harmonizes in a curious way with Fresnel's hypothesis that the velocity of longitudinal waves in ​the aether is indefinitely great compared with that of the transverse waves; for it is found by experiment with actual substances that the ratio of the velocity of propagation of longitudinal waves to that of transverse waves increases. rapidly as the medium becomes softer and more plastic.
In attempting to set forth a parallel between light and the vibrations of an elastic substance, the investigator is compelled more than once to make a choice between alternatives. He may, for instance, suppose that the vibrations of the aether are executed either parallel to the plane of polarization of the light. or at right angles to it; and he may suppose that the different refractive powers of different media are due either to differences in the inertia of the aether within the media, or to differences in its power of resisting distortion; or to both these causes combined There are, moreover, several distinct methods for avoiding the difficulties caused by the presence of longitudinal vibrations; and as, alas we shall see, a further source of diversity is to be found in that liability to error from which no man is free. It is therefore not surprising that the list of elastic-solid theories is a long one.
At the time when the transversality of light was discovered, no general method had been developed for investigating mathematically the properties of elastic bodies; but under the stimulus of Fresnel's discoveries, some of the best intellects of the age were attracted to the subject. The volume of Memoirs of the Academy which contains Fresnel's theory of crystal-optics contains also a memoir by Claud Louis Mario Henri Navier (b. 1785, d. 1836), at that time Professor of Mechanics in Paris, in which the correct equations of vibratory motion for a particular type of elastic solid were for the first time given. Navier supposed the medium to be ultimately constituted of an immense number of particles, which act on each other with forces directed along the lines joining them, and depending on their distances apart; and showed that if e denote. ​the (vector) displacement of the particle whose undisturbed position is (x, y, z), and if p denote the density of the medium, the equation of motion is
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-3n\ \mathrm {grad\ div} \ \mathbf {e} -n\ \mathrm {curl\ curl} \ \mathbf {e}$,
where n denotes a constant which measures the rigidity, or power of resisting distortion, of the mediun, All such elastic properties of the body as the velocity of propagation of waves in it must evidently depend on the ratio n/ρ.
Among the referees of one of Navier's papers was Augustine Louis Cauchy (b. 1789, d. 1857), one of the greatest analysts of the nineteenth century, who, becoming interested in the question, published in 1828 a discussion of it from an entirely different point of view. Instead of assuming, as Navier had done, that the medium is an aggregate of point-centres of force, and thus involving himself in doubtful molecular hypotheses, he devised a method of directly studying the elastic properties of matter in bulk, and by its means showed that the vibrations of an isotropic solid are determined by the equation
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-\left(k+{\frac {4}{3}}n\right)\mathrm {grad\ div} \ \mathbf {e} -n\ \mathrm {curl\ curl} \ \mathbf {e}$;
here n denotes, as before, the constant of rigidity, and the constant k, which is called the modulus of compression, denotes the ratio of a pressure to the cubical compression produced by it. Cauchy's equation evidently differs from Navier's in that ​two constants, k and n, appear instead of one. The reason for this is that a body constituted from point-centres of force in Navier's fashion has its moduli of rigidity and compression connected by the relation
$k={\frac {5}{3}}n$.
Actual bodies do not necessarily obey this condition; e.g. for india-rubber, k is much larger than ${\frac {5}{3}}n$; and there seems to be no reason why we should impose it on the aether.
In the same year Poisson succeeded in solving the differential equation which had thus been shown to determine the wave-motions possible in an elastic solid. The solution, which is both simple and elegant, may be derived as follows:—Let the displacement vector e be resolved into two components, of which one c is circuital, or satisfies the condition
div c = 0,
while the other b is irrotational, or satisfies the condition
curl b = 0.
The equation takes the form
$\rho {\frac {\partial ^{2}}{\partial t^{2}}}(\mathbf {b} +\mathbf {c} )-n\nabla ^{2}\mathbf {c} -\left(k+{\frac {4}{3n}}\right)\nabla ^{2}\mathbf {b} =0$.
​The terms which involve b and those which involve e must be separately zero, since they represent respectively the irrotational and the circuital parts of the equation. Thus, c satisfies the pair of equations
$\rho {\frac {\partial ^{2}\mathbf {c} }{\partial t^{2}}}=n\nabla ^{2}\mathbf {c}$, $\mathrm {div} \ \mathbf {c} =0$;
while b is to be determined from
$\rho {\frac {\partial ^{2}\mathbf {b} }{\partial t^{2}}}=\left(k+{\frac {4}{3}}n\right)\nabla ^{2}\mathbf {b}$, $\mathrm {curl} \ \mathbf {b} =0$.
A particular solution of the equations for c is easily seen to be
$c_{x}=A\sin \lambda \left(z-t{\sqrt {\frac {n}{\rho }}}\right)$, $c_{y}=B\sin \lambda \left(z-t{\sqrt {\frac {n}{\rho }}}\right)$, $c_{z}=0$,
which represents a transverse plane wave propagated with velocity ✓(n/ρ). It can be shown that the general solution of the differential equations for c is formed of such waves as this, travelling in all directions, superposed on each other.
A particular solution of the equations for b is
$b_{x}=0$, $b_{y}=0$, $b_{z}=C\sin \lambda \left(z-t{\sqrt {\frac {k+{\tfrac {4}{3}}n}{\rho }}}\right)$,
which represents a longitudinal wave propagated with velocity
${\sqrt {(k+{\tfrac {4}{3}}n)/\rho }}$;
the general solution of the differential equation for b is formed by the superposition of such waves as this, travelling in all directions.
Poisson thus discovered that the waves in an elastic solid are of two kinds: those in c are transverse, and are propagated with velocity (n/ρ)$^{1/2}$; while those in b are longitudinal, and are propagated with velocity {(k + 4/3n)/ρ}$^{1/2}$ The latter are waves of dilatation and condensation, like sound-waves; in the c-waves, on the other hand, the medium is not dilated or condensed, but ​only distorted in a manner consistent with the preservation of a constant density.
The researches which have been mentioned hitherto have all been concerned with isotropic bodies. Cauchy in 1828 extended the equations to the case of crystalline substances. This, however, he accomplished only by reverting to Navier's plan of conceiving an elastic body as a cluster of particles which attract each other with forces depending on their distances apart; the aelotropy he accounted for by supposing the particles to be packed more closely in some directions than in others.
The general equations thus obtained for the vibrations of an elastic solid contain twenty-one constants; six of those depend on the initial stress, so that if the body is initially without stress, only fifteen constants are involved. If, retaining the initial stress, the medium is supposed to be symmetrical with respect to three mutually orthogonal planes, the twenty-one constants reduce to nine, and the equations which determine the vibrations may be written in the form
* ${\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=(a+G){\frac {\partial ^{2}e_{x}}{\partial x^{2}}}+(h+H){\frac {\partial ^{2}e_{x}}{\partial y^{2}}}+(g+I){\frac {\partial ^{2}e_{x}}{\partial z^{2}}}$
* $+2{\frac {\partial }{\partial x}}\left(a{\frac {\partial e_{x}}{\partial x}}+h{\frac {\partial e_{y}}{\partial y}}+g{\frac {\partial e_{z}}{\partial z}}\right)$,
and two similar equations. The three constants G, H, I represent the stresses across planes parallel to the coordinate planes in the undisturbed state of the aether. ​On the basis of these equations, Cauchy worked out a theory of light, of which an instalment relating to crystal-optics was presented to the Academy in 1830. Its characteristic features will now be sketched.
By substitution in the equations last given, it is found that when the wave-front of the vibration is parallel to the plane of yz, the velocity of propagation must be (h + G)$^{1/2}$ if the vibration takes place parallel to the axis of y, and (g + G)$^{1/2}$ if it takes place parallel to the axis of z. Similarly when the wave-front is parallel to the plane of zx, the velocity must be (h + H)$^{1/2}$ if the vibration is parallel to the axis of x, and (f+ H)$^{1/2}$: if it is parallel to the axis of z; and when the wave-front is parallel to the plane xy, the velocity must be (g + I)$^{1/2}$ if the vibration is parallel to the axis of x, and (f + I)$^{1/2}$ if it is parallel to the axis of y.
Now it is known from experiment that the velocity of a ray polarized parallel to one of the planes in question is the same, whether its direction of propagation is along one or the other of the axes in that plane: so, if we assume that the vibrations which constitute light are executed parallel to the plane of polarization, we must have
$f+H=f+I$, $g+I=g+G$, $h+H=h+G$;
or, $G=H=I$. This is the assumption made in the memoir of 1830: the theory based on it is generally known as Cauchy's First Theory; the equilibrium pressures G, H, I, being all equal, are taken to be zero.
If, on the other hand, we make the alternative assumption that the vibrations of the aether are executed at right angles to the plane of polarization, we must have $h+H=4+1$, $f+I=h+G$, $g+G=f+H$; ​the theory based on this supposition is known as Cauchy's Second Theory: it was published in 1836.
In both theories, Cauchy imposes the condition that the section of two of the sheets of the wave-surface made by any one of the coordinate planes is to be formed of a circle and an ellipse, as in Fresnel's theory; this yields the three conditions
$3bc=f(b+c+f);3ba=g(c+a+g);3ab=h(a+b+h)$.
Thus in the first theory we have these together with the equations
$G=0,H=0,I=0$,
which express the condition that the undisturbed state of the aether is unstressed; and the aethereal vibrations are executed parallel to the plane of polarization. In the second theory we have the three first equations, together with
$f-G=h-I=g-H$;
and the plane of polarization is interpreted to be the plane at right angles to the direction of vibration of the aether.
Either of Cauchy's theories accounts tolerably well for the phenomena of crystal-optics; but the wave-surface (or rather the two sheets of it which correspond to nearly transverse waves) is not exactly Fresnel's. In both theories the existence of a third wave, formed of nearly longitudinal vibrations, is a formidable difficulty. Cauchy himself anticipated that the existence of these vibrations would ultimately be demonstrated by experiment, and in one place conjectured that they might be of a calorific nature. A further objection to Cauchy's theories is that the relations between the constants do not appear to admit of any simple physical interpretation, being evidently assumed for the sole purpose of forcing the formulae into some degree of conformity with the results of experiment. And further difficulties will appear when we proceed subsequently to compare the properties which are assigned to the aether in crystal-optics with those which must be postulated in order to account for reflexion and refraction.
​To the latter problem Cauchy soon addressed himself, his investigations being in fact published in the same year (1830) as the first of his theories of crystal-optics.
At the outset of any work on refraction, it is necessary to assign a cause for the existence of refractive indices, i.e. for the variation in the velocity of light from one body to another. Huygens, as we have seen, suggested that transparent bodies consist of hard particles which interact with the aethereal matter, modifying its elasticity. Cauchy in his earlier papers followed this lead more or less closely, assuming that the density ρ of the aether is the same in all media, but that 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, and the axis of z parallel to the intersection of this interface with the incident wave-front; and suppose, first, that the incident vibration is executed at right angles to the plane of incidence, so that it may be represented by
$e_{z}=f\left(-x\cos \ i-y\sin \ i+{\sqrt {\frac {n}{\rho }}}t\right)$,
where i denotes the angle of incidence; the reflected wave may be represented by
$e_{z}=F\left(x\cos \ i-y\sin \ i+{\sqrt {\frac {n}{\rho }}}t\right)$,
and the refracted wave by
$e_{z}=f_{1}\left(-x\cos \ r-y\sin \ r+{\sqrt {\frac {n^{\prime }}{\rho }}}t\right)$,
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
$n{\frac {\partial e_{z}}{\partial x}}$ and $n{\frac {\partial e_{z}}{\partial y}}$
are to be continuous across the interface: so we have
$\textstyle n\cos i^{\prime }.(f^{\prime }-F^{\prime })=n^{\prime }\cos r.f_{1}^{\prime }$; $n\sin i.(f^{\prime }+F^{\prime })=n^{\prime }\sin r.f_{1}^{\prime }$.
Eliminating f′$_1$, we have
${\frac {F^{\prime }}{f^{\prime }}}={\frac {\sin(r-i)}{\sin(r+i)}}$
Now this is Fresnel's sine-law for the ratio of the intensity of the reflected ray to that of the incident ray; and it is known that the light to which it applies is that which is polarized parallel to the plane of incidence. Thus Cauchy was driven to the conclusion 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.
The case of a vibration performed in the plane of incidence he discussed in the same way. It was found that Fresnel's tangent-law could be obtained by assuming that e$_x$ and the normal pressure across the interface have equal values in the two contiguous media.
The theory thus advanced was encumbered with many difficulties. In the first place, 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. In the second place, no reasons were given for the choice of the conditions at the interface. Cauchy's motive in selecting these particular conditions was evidently to secure the fulfilment of 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 the decision, which Cauchy now made, 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 made another attempt to frame a satisfactory theory of reflexion, based on the assumption just mentioned, and on the following boundary-conditions:—At the interface between two media curl e is to be continuous, and (taking the axis of x normal to the interface) $\partial e_{x}/\partial x$ is also to be continuous.
Again we find no very satisfactory reasons assigned for the choice of the boundary-conditions, and as 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. 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.
It may be remarked that 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 ten years the great French mathematician produced two distinct theories of crystal-optics and three distinct theories of reflexion, 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.
Cauchy's theories, then, 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 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; it is obvious that 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 three components of the displacement must be continuous at the interface, and 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 three true boundary-conditions, however, they used only a single equation, 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, of course, the same device as had been used previously by Fresnel; it must, however, be remarked that 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. Indeed, 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 read to the Cambridge Philosophical Society in December, 1837. Green, though inferior to Cauchy as an analyst, was his superior in physical insight; instead of designing boundary-equations for the express purpose of yielding Fresnel's sine and tangent formulae, he set to work to determine the conditions which are actually satisfied at the interfaces of real elastic solids, These he obtained by means of general dynamical principles. In an isotropic medium which is strained, the potential energy per unit volume due to the state of stress is
* $\phi ={\frac {1}{2}}\left(k+{\frac {4}{3}}n\right)\left({\frac {\partial e_{x}}{\partial x}}+{\frac {\partial e_{y}}{\partial y}}+{\frac {\partial e_{z}}{\partial z}}\right)^{2}+{\frac {1}{2}}n\left\{\left({\frac {\partial e_{z}}{\partial y}}+{\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{x}}{\partial z}}+{\frac {\partial e_{z}}{\partial x}}\right)^{2}\right.$
* $\left.+\left({\frac {\partial e_{y}}{\partial x}}+{\frac {\partial e_{x}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\$,
where e denotes the displacement, and k and n denote the two ​elastic constants already introduced; by substituting this value of φ in the general variational equation
$\iiint \rho \left\{{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}\partial e_{x}+{\frac {\partial ^{2}e_{y}}{\partial t^{2}}}\partial e_{x}y+{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}\partial e_{z}\right\}\ dx\ dy\ dz=-\iiint \partial \phi \ dx\ dy\ dz$
(where ρ denotes the density), the equation of motion may be deduced.
But this method does more than merely furnish the equation of motion
$\rho {\ddot {\mathbf {e} }}=-\left(k+{\frac {4}{3}}n\right)\ \mathrm {grad\ div} \ \mathbf {e} -n\ \mathrm {curl\ curl} \ \mathbf {e}$;
or,
$\rho {\ddot {\mathbf {e} }}=-\left(k+{\frac {1}{3}}n\right)\mathrm {grad\ div} \ \mathbf {e} +n\nabla ^{2}\mathbf {e}$,
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 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 three quantities are
$\left(k-{\frac {2}{3}}n\right)\mathrm {div} \ \mathbf {e} +2n{\frac {\partial e_{x}}{\partial _{x}}}$, $n\left({\frac {\partial e_{z}}{\partial x}}+{\frac {\partial e_{x}}{\partial z}}\right)$, and $n\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)$.
The correct boundary-conditions being thus obtained, it was a simple matter to discuss the reflexion and refraction of an incident wave by the procedure of Fresnel and Cauchy. The result found by Green was that if the vibration of the aethereal molecules is executed at right angles to the plane of incidence, the intensity of the reflected light obeys Fresnel's sine-law, provided the rigidity n is assumed to be the same for all media, but the inertia ρ to vary from one medium to another. Since the sine-law is known to be true for light polarized in the plane of incidence, Green's conclusion confirmed the hypotheses of ​Fresnel, that the vibrations are executed at right angles to the plane of polarization, and that the optical differences between media are due to the different densities of aether within them.
It now remained for Green to discuss the case in which the incident light is polarized at right angles to the plane of incidence, so that the motion of the acthereal particles is parallel to the intersection of the plane of incidence with the front of the In this case it is impossible to satisfy all the six boundary-conditions without assuming that longitudinal vibrations are generated by the act of reflexion. Taking the plane of incidence to be the plane of yz, and the interface to be the plane of xy, the incident wave may be represented by the equations
$e_{y}=A{\frac {\partial }{\partial z}}f(t+lz+my)$; $e_{z}=-A{\frac {\partial }{\partial y}}f(t+lz+my)$;
where, if i deote the angle of incidence, we have
$l={\sqrt {\frac {\rho _{1}}{n}}}\cos i$, $m=-{\sqrt {\frac {\rho _{1}}{n}}}\sin i$.
There will be a transverse reflected wave,
$e_{y}=B{\frac {\partial }{\partial z}}f(t-lz+my)$; $e_{z}=-B{\frac {\partial }{\partial y}}f(t-lz+my)$;
and a transverse refracted wave,
$e_{y}=C{\frac {\partial }{\partial z}}f(t+l_{1}z+my)$; $e_{z}=-C{\frac {\partial }{\partial y}}f(t+l_{1}z+my)$,
where, since the velocity of transverse waves in the second medium is $\textstyle {\sqrt {n/\rho _{2}}}$, we can determine l$_1$ from the equation
$l_{1}^{2}+m^{2}={\frac {\rho _{2}}{n}}$;
there will also be a longitudinal reflected wave,
$e_{y}=D{\frac {\partial }{\partial y}}f(t-\lambda z+my)$; $e_{z}=D{\frac {\partial }{\partial z}}f(t-\lambda z+my)$,
$\lambda ^{2}+m^{2}={\frac {\rho _{1}}{k_{1}+{\tfrac {4}{3}}n}}$;
and a longitudinal refracted wave,
$e_{y}=E{\frac {\partial }{\partial y}}f(t+\lambda _{1}z+my)$; $e_{z}=E{\frac {\partial }{\partial z}}f(t+\lambda _{1}z+my)$,
where λ$_1$, is determined by
$\lambda _{1}^{2}+m^{2}={\frac {\rho _{2}}{k_{2}+{\tfrac {4}{3}}n}}$.
Substituting these values for the displacement in the boundary-conditions which have been already formulated, we obtain the equations which determine the intensities of the reflected and refracted waves; in particular, it appears that the amplitude of the reflected transverse wave is given by the equation
${\frac {A-B}{A+B}}={\frac {l_{1}\rho _{1}}{l\rho _{2}}}+{\frac {m^{2}}{l}}{\frac {(\rho _{1}-\rho _{2})^{2}}{\rho _{2}(\lambda \rho _{2}+\lambda _{1}\rho _{1})}}$.
Now if the elastic constants of the media are such that the velocities of propagation of the longitudinal waves are of the same order of magnitude as those of the transverse waves, the direction-cosines of the longitudinal reflected and refracted rays will in general have real values, and these rays will carry away some of the energy which is brought to the interface by the incident wave. Green avoided this difficulty by adopting Fresnel's suggestion that the resistance of the aether to compression may be very large in comparison with the resistance to distortion, as is actually the case with such substances as jelly and caoutchouc: in this case the longitudinal waves are degraded in much the same way as the transverse refracted ray is degraded when there is total reflexion, and so do not carry away energy. Making this supposition, so that k$_1$ and k$_2$ are very large, the quantities λ and λ$_1$, have the values m$\textstyle {\sqrt {-1}}$, and we have
${\frac {A-B}{A+B}}={\frac {l_{1}\rho _{1}}{l\rho _{2}}}+{\frac {m^{2}}{l}}{\frac {(\rho _{1}-\rho _{2})^{2}}{\rho _{2}(\rho _{1}+\rho _{2})}}{\sqrt {-1}}$.
$\textstyle {\bar {B}}/{\bar {A}}$
$\left({\frac {\bar {B}}{\bar {A}}}\right)^{2}={\frac {\left({\frac {\rho _{2}}{\rho _{1}}}+1\right)^{2}\left({\frac {\rho _{2}}{\rho _{1}}}-{\frac {l_{1}}{l}}\right)^{2}+\left({\frac {\rho _{2}}{\rho _{1}}}-1\right)^{4}{\frac {m^{2}}{l^{2}}}}{\left({\frac {\rho _{2}}{\rho _{1}}}+1\right)^{2}\left({\frac {\rho _{2}}{\rho _{1}}}+{\frac {l_{1}}{l}}\right)^{2}+\left({\frac {\rho _{2}}{\rho _{1}}}-1\right)^{4}{\frac {m^{2}}{l^{2}}}}}$
This expression represents the ratio of the intensity of the transverse reflected wave to that the incident wave. It does not agree with Fresnel's tangent-formula: and both on this account and also because (as we shall see) this theory of reflexion does not harmonize well with the clastic-solid theory of crystal-optics, it must be concluded that the vibrations of a Greenian solid do not furnish an exact parallel to the vibrations which constitute light.
The success of Green's investigation from the standpoint of dynamics, set off by its failure in the details last mentioned, stimulated MacCullagh to fresh exertions. At length he succeeded in placing his own theory, which had all along been free from reproach so far as agreement with optical experiments was concerned, on a sound dynamical basis; thereby effecting that reconciliation of the theories of Light and Dynamics which had been the dream of every physicist since the days of Descartes.
The central feature of MacCullagh's investigation, which was presented to the Royal Irish Academy in 1839, is the introduction of a new type of elastic solid. He had, in fact, concluded from Green's results that it was impossible to explain optical phenomena satisfactorily by comparing the aether to an elastic solid of the ordinary type, which resists compression, and distortion; and he saw that the only hope of the situation was to devise a medium which should be as strictly conformable to dynamical laws as Green's elastic solid, and yet should have its properties specially designed to fulfil the requirements of the theory of light. Such a medium he now described.
If as before we denote by e the vector displacement of a point of the medium from its equilibrium position, it is well ​known that the vector curl e denotes twice the rotation of the part of the solid in the neighbourhood of the point (x, y, z) from its equilibrium orientation. In an ordinary elastic solid, the potential energy of strain depends only on the change of size and shape of the volume-elements; on their compression and distortion, in fact. For MacCullagh's new medium, on the other hand, the potential energy depends only on the rotation of the volume-elements.
Since the medium is not supposed to be in a state of stress in its undisturbed condition, the potential energy per unit volume must be a quadratic function of the derivates of e; so that in an isotropic medium this quantity φ must be formed from the only in variant which depends solely on the rotation and is quadratic in the derivates, that is from (curl e)$^2$; thus we may write
$\phi ={\tfrac {1}{2}}\left\{\left({\frac {\partial e_{z}}{\partial y}}-{\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{x}}{\partial z}}-{\frac {\partial e_{z}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial x}}-{\frac {\partial e_{x}}{\partial y}}\right)^{2}\right\}$.
The equation of motion is now to be determined, as in the case of Green's aether, from the variational equation
$\iiint \rho \left\{{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}\delta e_{x}{\frac {\partial ^{2}e_{y}}{\partial t^{2}}}\delta e_{y}{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}\delta e_{z}\right\}\ dx\ dy\ dz=\iiint \delta \phi \ dx\ dy\ dz$;
the result is
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-\mu {\text{ curl curl }}\mathbf {e}$.
It is evident from this equation that if div e is initially zero it will always be zero: we shall suppose this to be the case, so that no longitudinal waves exist at any time in the medium. One of the greatest difficulties which beset elastic-solid theories is thus completely removed.
The equation of motion may now be written
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=\mu \nabla ^{2}\mathbf {e}$,
$\textstyle {\sqrt {\mu /\rho }}$.
From the variational equation we may also determine the boundary-conditions which must be satisfied at the interface between two media; these are, that the three components of e are to be continuous across the interface, and that the two components of curl e parallel to the interface are also to be continuous across it. One of these five conditions, namely, the continuity of the normal component of e, is really dependent on the other four; for if we take the axis of x normal to the interface, the equation of motion gives
$\rho {\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=-{\frac {\partial }{\partial y}}(\mu {\text{ curl }}\mathbf {e} )_{z}+{\frac {\partial }{\partial z}}(\mu {\text{ curl }}\mathbf {e} )_{y}$
and as the quantities ρ, (μ curl e)$_z$ and (μ curl e)$_y$, are continuous across the interface, the continuity of ∂$^2$e$_x$/∂t$^2$ follows. Thus the only independent boundary-conditions in MacCullagh's theory are the continuity of the tangential components of e and of μ curl e. It is easily seen that these are equivalent to the boundary-conditions used in MacCullagh's earlier paper, namely, the equation of vis viva and the continuity of the three components of e: and thus the "rotationally elastic" aether of this memoir furnishes a dynamical foundation for the memoir of 1837.
The extension to crystalline media is made by assuming the potential energy per unit volume to have, when referred to the principal axes, the form
$A\left({\frac {\partial e_{z}}{\partial y}}-{\frac {\partial e_{y}}{\partial z}}\right)^{2}+B\left({\frac {\partial e_{x}}{\partial z}}-{\frac {\partial e_{z}}{\partial x}}\right)^{2}+C\left({\frac {\partial e_{y}}{\partial x}}-{\frac {\partial e_{x}}{\partial y}}\right)^{2}$
where A, B, C denote three constants which determine the optical behaviour of the medium: it is readily seen that the wave-surface is Fresnel's, and that the plane of polarization ​contains the displacement, and is at right angles to the rotation.
MacCullagh's work was regarded with doubt by his own and the succeeding generation of mathematical physicists, and can scarcely be said to have been properly appreciated until FitzGerald drew attention to it forty years afterwards. But. there can be no doubt that MacCullagh really solved the problem of devising a medium whose vibrations, calculated in accordance with the correct laws of dynamics, should have the same properties as the vibrations of light.
The hesitation which was felt in accepting the rotationally elastic aether arose mainly from the want of any readily conceived example of a body endowed with such a property. This difficulty was removed in 1889 by Sir William Thomson (Lord Kelvin), who designed mechanical models possessed of rotational elasticity. Suppose, for example, that a structure is. formed of spheres, each sphere being in the centre of the tetrahedron formed by its four nearest neighbours. Let each sphere be joined to these four neighbours by rigid bars, which have spherical caps at their ends so as to slide freely on the spheres. Such a structure would, for small deformations, behave like an incompressible perfect fluid. Now attach to each bar a. pair of gyroscopically-mounted flywheels, rotating with equal and opposite angular velocities, and having their axes in the line of the bar: a bar thus equipped will require a couple to hold it at rest in any position inclined to its original position, and the structure as a whole will possess that kind of quasi-elasticity which was first imagined by MacCullagh.
This particular representation is not perfect, since a system of forces would be required to hold the model in equilibrium if it were irrotationally distorted. Lord Kelvin subsequently invented another structure free from this defect.
​The work of Green proved a stimulus not only to MacCullagh but to Cauchy, who now (1839) published yet a third theory of reflexion. This appears to have owed its origin to a remark of Green's, that the longitudinal wave might be avoided in either of two ways—namely, by supposing its velocity to be indefinitely great or indefinitely small. Green curtly dismissed the latter alternative and adopted the former, on the ground that the equilibrium of the medium would be unstable if its compressibility were negative (as it must be if the velocity of longitudinal waves is to vanish). Cauchy, without attempting to meet Green's objection, took up the study of a medium whose elastic constants are connected by the equation
$k+{\frac {4}{3}}=0$,
so that the longitudinal vibrations have zero velocity; and showed that if the aethereal vibrations are supposed to be executed at right angles to the plane of polarization, and if the rigidity of the aether is assumed to be the same in all media, a ray which is reflected will obey the sine-law and tangent-law of Fresnel. The boundary-conditions which he adopted in order to obtain this result were the continuity of the displacement e and of its derivate $\partial \mathbf {e} /\partial x$, where the axis of x is taken at right angles to the interface. These are not the true boundary-conditions for general elastic solids; but in the particular case now under discussion, where the rigidity is the same in the two media, they yield the same equations as the conditions correctly given by Green.
The aether of Cauchy's third theory of reflexion is well worthy of some further study. It is generally known as the contractile or labile aether, the names being due to William ​Thomson (Lord Kelvin), who discussed it long afterwards. It may be defined as an elastic medium of (negative) compressibility such as to make the velocity of the longitudinal wave zero: this implies that no work is required to be done in order to give the medium any small irrotational disturbance. An example is furnished by homogeneous foam free from air and held from collapse by adhesion to a containing vessel.
Cauchy, as we have seen, did not attempt to refute Green's objection that such a medium would be unstable; but, as Thomson remarked, every possible infinitesimal motion of the medium is, in the elementary dynamics of the subject, proved to be resolvable into coexistent wave-motions. If, then, the velocity of propagation for each of the two kinds of wave-motion is real, the equilibrium must be stable, provided the medium either extends through boundless space or has a fixed containing vessel as its boundary.
When the rigidity of the luminiferous medium is supposed to have the same value in all bodies, the conditions to be satisfied at an interface reduce to the continuity of the displacement e, of the tangential components of curl e, and of the scalar quantity (k + 4/3n) div e across the interface.
Now we have seen that when a transverse wave is incident on an interface, it gives rise in general to reflected and refracted waves of both the transverse and the longitudinal species. In the case of the contractile aether, for which the velocity of propagation of the longitudinal waves is very small, the ordinary construction for refracted waves shows that the directions of propagation of the reflected and refracted longitudinal waves will be almost normal to the interface. The longitudinal waves will therefore contribute only to the component of displacement normal to the interface, not to the tangential components: in other words, the only tangential components of displacement at the interface are those due to the three transverse waves-the incident, reflected, and refracted. Moreover, the longitudinal waves do not contribute at all to curl e; and, ​therefore, in the contractile aether, the conditions that the tangential components of e and of n curl e shall be continuous across an interface are satisfied by the distortional part of the disturbance taken alone. The condition that the component of e normal to the interface is to be continuous is not satisfied by the distortional part of the disturbance taken alone, but is satisfied when the distortional and compressional parts are taken together.
The energy carried away by the longitudinal waves is infinitesimal, as might be expected, since no work is required in order to generate an irrotational displacement. Hence, with this aether, the behaviour of the transverse waves at an interface may be specified without considering the irrotational part of the disturbance at all, by the conditions that the conservation of energy is to hold and that the tangential components of e and of n curl e are to be continuous. But if we identify these transverse waves with light, assuming that the displacement e is at right angles to the plane of polarization of the light, and assuming moreover that the rigidity n is the same in all media (the differences between media depending on differences in the inertia ρ), we have exactly the assumptions of Fresnel's theory of light: whence it follows that transverse waves in the labile aether must obey in reflexion the sinc-law and tangent-law of Fresnel.
The great advantage of the labile aether is that it overcomes the difficulty about securing continuity of the normal component of displacement at an interface between two media: the light-waves taken alone do not satisfy this condition of continuity; but the total disturbance consisting of light-waves and irrotational disturbance taken together does satisfy it; and this is ensured without allowing the irrotational disturbance to carry off any of the energy.
​William Thomson (Lord Kelvin, b. 1824, d. 1908), who devoted much attention to the labile aether, was at one time led to doubt the validity of this explanation of light; for when investigating the radiation of energy from a vibrating rigid globe embedded in an infinite elastic-solid aether, he found that in some cases the irrotational waves would carry away a considerable part of the energy if the aether were of the labile type. This difficulty, however, was removed by the observation that it is sufficient for the fulfilment of Fresnel's laws if the velocity of the irrotational waves in one of the two media is very small, without regard to the other medium. Following up this idea, Thomson assumed that in space void of ponderable matter the aether is practically incompressible by the forces concerned in light-waves, but that in the space occupied by liquids and solids it has a negative compressibility, so as to give zero velocity for longitudinal aether-waves in these bodies. This assumption was based on the conception that material atoms move through space without displacing the aether: a conception which, as Thomson remarked, contradicts the old scholastic axiom that two different portions of matter cannot simultaneously occupy the same space. He supposed the aether to be attracted and repelled by the atoms, and thereby to be condensed or rarefied.
The year 1839, which saw the publication of MacCullagh's dynamical theory of light and Cauchy's theory of the labile aether, was memorable also for the appearance of a memoir by Green on crystal-optics. This really contains two distinct theories, which respectively resemble Cauchy's First and Second Theories: in one of them, the stresses in the undisturbed state ​of the aether are supposed to vanish, and the vibrations of the aether are supposed to be executed parallel to the plane of polarization of the light; in the other theory, the initial stresses are not supposed to vanish, and the aether-vibrations are at right angles to the plane of polarization. The two investigations are generally known as Green's First and Second 'Theories of crystal-optics.
The foundations of both theories are, however, the same. Green first of all determined the potential energy of a strained crystalline solid; this in the most general case involves 27 constants, or 21 if there is no initial stress. If, however, as is here assumed, the medium possesses the planes of symmetry at right angles to each other, the number of constants reduces to 12, or to 9 if there is no initial stress; if e denote the displacement, the potential energy per unit volume may be written
* $G{\frac {\partial e_{x}}{\partial x}}+H{\frac {\partial e_{y}}{\partial y}}+I{\frac {\partial e_{z}}{\partial z}}$
* ${\tfrac {1}{2}}G\left\{\left({\frac {\partial e_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial x}}\right)^{2}\right\}+{\tfrac {1}{2}}H\left\{\left({\frac {\partial e_{x}}{\partial y}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial y}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial y}}\right)^{2}\right\}$
* ${\tfrac {1}{2}}I\left\{\left({\frac {\partial e_{x}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial z}}\right)^{2}\right\}$
* $+{\tfrac {3}{2}}a\left({\frac {\partial e_{x}}{\partial x}}\right)^{2}+{\tfrac {3}{2}}b\left({\frac {\partial e_{y}}{\partial y}}\right)^{2}+{\tfrac {3}{2}}c\left({\frac {\partial e_{z}}{\partial z}}\right)^{2}$
* $+f^{\prime }{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}+g^{\prime }{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{z}}{\partial z}}+h^{\prime }{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}$
* $+{\tfrac {1}{2}}f\left({\frac {\partial e_{y}}{\partial z}}+{\frac {\partial e_{z}}{\partial y}}\right)^{2}+{\tfrac {1}{2}}g\left({\frac {\partial e_{x}}{\partial z}}+{\frac {\partial e_{z}}{\partial x}}\right)^{2}+{\tfrac {1}{2}}h\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)^{2}$.
The usual variational equation
$\iiint \rho \left\{{\frac {\partial ^{e}e_{x}}{\partial t^{2}}}\delta e_{x}+{\frac {\partial ^{e}e_{y}}{\partial t^{2}}}\delta e_{y}+{\frac {\partial ^{e}e_{z}}{\partial t^{2}}}\delta e_{z}\right\}\ dx\ dy\ dz=-\iiint \delta \phi \ dx\ dy\ dz$
* $\rho {\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=(a+G){\frac {\partial ^{2}e_{x}}{\partial x^{2}}}+(h+H){\frac {\partial ^{2}e_{x}}{\partial y^{2}}}+(g+I){\frac {\partial ^{2}e_{x}}{\partial z^{2}}}$
* $+{\frac {\partial }{\partial x}}\left(a{\frac {\partial e_{x}}{\partial x}}+h{\frac {\partial e_{y}}{\partial y}}+g{\frac {\partial e_{z}}{\partial z}}\right)+{\frac {\partial }{\partial x}}\left(a{\frac {\partial e_{x}}{\partial x}}+h^{\prime }{\frac {\partial e_{y}}{\partial y}}+g^{\prime }{\frac {\partial e_{z}}{\partial z}}\right)$
and two similar equations.
These differ from Cauchy's fundamental equations in having greater generality: for Cauchy's medium was supposed to be built up of point-centres of force attracting each other according to some function of the distance; and, as we have seen, there are limitations in this method of construction, which render it incompetent to represent the most general type of elastic solid. Cauchy's equations for crystalline media are, in fact, exactly analogous to the equations originally found by Navier for isotropic media, which contain only one elastic constant instead of two.
The number of constants in the above equations still exceeds the three which are required to specify the properties of a biaxal crystal: and Green now proceeds to consider how the number may be reduced. The condition which he imposes for this purpose is that for two of the three waves whose front is parallel to a given plane, the vibration of the aethereal molecules shall be accurately in the plane of the wave: in other words, that two of the three waves shall be purely distortional, the remaining one being consequently a normal vibration. This condition gives five relations, which may be written:—
$a=b=c={\tfrac {1}{3}}\mu$;
$f^{\prime }=\mu -2f$ $g^{\prime }=\mu -2g$ $h^{\prime }=\mu -2h$;
where μ denotes a new constant. ​Thus the potential energy per unit volume may be written
* $\phi =G{\frac {\partial e_{x}}{\partial x}}+H{\frac {\partial e_{y}}{\partial y}}+I{\frac {\partial e_{z}}{\partial z}}$
* $+{\tfrac {1}{2}}G\left\{\left({\frac {\partial e_{x}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial x}}\right)^{2}\right\}+{\tfrac {1}{2}}H\left\{\left({\frac {\partial e_{x}}{\partial y}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial y}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial y}}\right)^{2}\right\}\qquad$
* $+{\tfrac {1}{2}}I\left\{\left({\frac {\partial e_{x}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{z}}{\partial z}}\right)^{2}\right\}$
* $+{\tfrac {1}{2}}\mu \left({\frac {\partial e_{x}}{\partial x}}+{\frac {\partial e_{y}}{\partial y}}+{\frac {\partial e_{z}}{\partial z}}\right)^{2}$
* $+{\tfrac {1}{2}}f\left\{\left({\frac {\partial e_{y}}{\partial z}}+{\frac {\partial e_{z}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}\right\}+{\tfrac {1}{2}}g\left\{\left({\frac {\partial e_{z}}{\partial x}}+{\frac {\partial e_{x}}{\partial z}}\right)^{2}-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}\right\}$
* $+{\tfrac {1}{2}}h\left\{\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)^{2}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\}$.
At this point Green's two theories of crystal-optics diverge from each other. According to the first theory, the initial stresses G, H, I are zero, so that
* ${\tfrac {1}{2}}\mu \left({\frac {\partial e_{x}}{\partial x}}+{\frac {\partial e_{y}}{\partial y}}+{\frac {\partial e_{z}}{\partial z}}\right)^{2}$
* $+{\tfrac {1}{2}}f\left\{\left({\frac {\partial e_{y}}{\partial z}}+{\frac {\partial e_{z}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}\right\}+{\tfrac {1}{2}}g\left\{\left({\frac {\partial e_{z}}{\partial x}}+{\frac {\partial e_{x}}{\partial z}}\right)^{2}-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}\right\}\qquad$
* $+{\tfrac {1}{2}}h\left\{\left({\frac {\partial e_{x}}{\partial y}}+{\frac {\partial e_{y}}{\partial x}}\right)^{2}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\}$.
​This expression contains the correct number of constants, namely, four: three of them represent the optical constants of a biaxal crystal, and one (namely, μ) represents the square of the velocity of propagation of longitudinal waves. It is found that the two sheets of the wave-surface which correspond to the two distortional waves form a Fresnel's wave-surface, the third sheet, which corresponds to the longitudinal wave, being an ellipsoid. The directions of polarization and the wave-velocities of the distortional waves are identical with those assigned by Fresnel, provided it is assumed that the direction of vibration of the aether-particles is parallel to the plane of polarization; but this last assumption is of course inconsistent with Green's theory of reflexion and refraction.
In his Second Theory, Green, like Cauchy, used the condition that for the waves whose fronts are parallel to the coordinate planes, the wave-velocity depends only on the plane of polarization, and not on the direction of propagation. He thus obtained the equations already found by Cauchy—
G - f = H - g = I - h.
The wave-surface in this case also is Fresnel's, provided it is assumed that the vibrations of the aether are executed at right angles to the plane of polarization.
The principle which underlies the Second Theories of Green and Cauchy is that the aether in a crystal resembles an elastic solid which is unequally pressed or pulled in different directions by the unmoved ponderable matter. This idea appealed strongly to W. Thomson (Kelvin), who long afterwards developed it further, arriving at the following interesting result:—Let an incompressible solid, isotropic when unstrained, be such that its potential energy per unit volume is
${\frac {1}{2}}q\left({\frac {1}{\alpha }}+{\frac {1}{\beta }}+{\frac {1}{\gamma }}-3\right)$,
where q denotes its modulus of rigidity when unstrained, and ​α$^{1/2}$, β$^{1/2}$, γ$^{1/2}$, denote the proportions in which lines parallel to the axes of strain are altered; then if the solid be initially strained in a way defined by given values of α, β, γ, by forces applied to its surface, and if waves of distortion be superposed on this initial strain, the transmission of these waves will follow exactly the laws of Fresnel's theory of crystal-optics, the wave-surface being
${\frac {x^{2}}{{\frac {\alpha }{q}}r^{2}-1}}+{\frac {y^{2}}{{\frac {\beta }{q}}r^{2}-1}}+{\frac {z^{2}}{{\frac {\gamma }{q}}r^{2}-1}}=0$.
There is some difficulty in picturing the manner in which the molecules of ponderable matter act upon the aether so as to produce the initial strain required by this theory. Lord Kelvin utilized the suggestion to which we have already referred, namely, that the aether may pervade the atoms of matter so as to occupy space jointly with them, and that its interaction with them may consist in attractions and repulsions exercised throughout the regions interior to the atoms. These forces may be supposed to be so large in comparison with those called into play in free aether that the resistance to compression may be overcome, and 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, "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; and, indeed, 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.
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. 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, 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 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.
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.
​A way of escape from this conclusion suggested itself to Stokes, and later to Rankine and Lord Rayleigh. 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
$\rho _{1}\left({\frac {\partial e_{x}}{\partial t}}\right)^{2}+\rho _{2}\left({\frac {\partial e_{y}}{\partial t}}\right)^{2}+\rho _{3}\left({\frac {\partial e_{z}}{\partial t}}\right)^{2}$
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
* ${\tfrac {1}{2}}n\left\{\left({\frac {\partial e_{z}}{\partial y}}+{\frac {\partial e_{y}}{\partial z}}\right)^{2}+\left({\frac {\partial e_{x}}{\partial z}}+{\frac {\partial e_{z}}{\partial x}}\right)^{2}+\left({\frac {\partial e_{y}}{\partial x}}+{\frac {\partial e_{x}}{\partial y}}\right)^{2}-4{\frac {\partial e_{y}}{\partial y}}{\frac {\partial e_{z}}{\partial z}}\right.$
* $\left.-4{\frac {\partial e_{z}}{\partial z}}{\frac {\partial e_{x}}{\partial x}}-4{\frac {\partial e_{x}}{\partial x}}{\frac {\partial e_{y}}{\partial y}}\right\$,
where n denotes as usual the rigidity. ​The variational equation of motion is
* $\iiint \left\{\rho _{1}{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}\delta e_{x}+\rho _{1}{\frac {\partial ^{2}e_{y}}{\partial t^{2}}}\delta e_{y}+\rho _{1}{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}\delta e_{z}\right\}\ dx\ dy\ dz$
* $=-\iiint \left\{\delta \phi -p\delta \left({\frac {\partial e_{x}}{\partial x}}+{\frac {\partial e_{y}}{\partial y}}+{\frac {\partial e_{z}}{\partial z}}\right)\right\}\ dx\ dy\ dz$
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
${\text{div }}\mathbf {e} =0$
The equations of motion which result from this variational equation are
$\rho _{1}{\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=-{\frac {\partial p}{\partial x}}+n\nabla ^{2}e_{x}$
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
${\frac {\lambda ^{2}}{n-\rho _{1}V^{2}}}+{\frac {\mu ^{2}}{n-\rho _{2}V^{2}}}+{\frac {\nu ^{2}}{n-\rho _{3}V^{2}}}=0$
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. About 1867 Stokes carried out a series of experiments in order to determine which of the two theories was most nearly conformable to the facts: he found the construction of Huygens and Fresnel to be decidedly the more correct, the difference between the results of it and the rival construction being about 100 times the probable error of observation.
​The hypothesis that in crystals the inertia depends on direction seemed therefore to be discredited when the theory based on it was compared with the results of observation. But when, in 1888, W. Thomson (Lord Kelvin) revived Cauchy's. theory of the labile aether, the question naturally arose as to whether that theory could be extended so as to account for the optical properties of crystals: and it was shown by R. T. Glazebrook that the correct formulae of crystal-optics are obtained when the Cauchy-Thomson hypothesis of zero velocity for the longitudinal wave is combined with the Stokes-Rankine-Rayleigh hypothesis of aelotropic inertia.
For on reference to the formulae which have been already given, it is obvious that the equation of motion of an aether having these properties must be
$(\rho _{1}{\overset {..}{e}}_{x},\rho _{2}{\overset {..}{e}}_{y},\rho _{3}{\overset {..}{e}}_{z})=-n\mathrm {curl} \ \mathrm {curl} \ \mathbf {e}$,
where e denotes the displacement, n the rigidity, and (ρ$_1$, ρ$_2$, ρ$_3$) the inertia: and this equation leads by the usual analysis to Fresnel's wave-surface. The displacement e of the aethereal particles is not, however, accurately in the wave-front, as in Fresnel's theory, but is at right angles to the direction of the ray, in the plane passing through the ray and the wave-normal.
Having now traced the progress of the elastic-solid theory so far as it is concerned with the propagation of light in ordinary isotropic media and in crystals, we must consider the attempts which were made about this time to account for the optical properties of a more peculiar class of substances.
It was found by Arago in 1811 that the state of polarization of a beam of light is altered when the beam is passed through a plate of quartz along the optic axis. The ​phenomenon was studied shortly afterwards by Biot, who showed that the alteration consists in a rotation of the plane of polarization about the direction of propagation: the angle of rotation is proportional to the thickness of the plate and inversely proportional to the square of the wave-length.
In some specimens of quartz the rotation is from left to right, in others from right to left. This distinction was shown by Sir John Herschel (b. 1792, d. 1871) in 1820 to be associated with differences in the crystalline forn of the specimens, the two types bearing the same relation to each other as a right-handed and left-handed helix respectively. Fresnel and W. Thomson proposed the term helical to denoto the property of rotating the plane of polarization, exhibited by such bodies as quartz: the less appropriato term natural rotatory polarization is, however, generally used.
Biot showed that many liquid organic bodies, e.g. turpentine and sugar solutions, possess the natural rotatory property: we might be led to infer the presence of a helical structure in the molecules of such substances; and this inference is supported by the study of their chemical constitution; for they are invariably of the mirror-image' or "enantiomorphous" type, in which one of the atoms (generally carbon) is asymmetrically linked to other atoms.
The next advance in the subject was due to Fresnel, who showed that in naturally active bodies the velocity of propagation of circularly polarized light is different according as the polarization is right-handed or left-handed. From this property the rotation of the plane of polarization of a plane. polarized ray may be immediately deduced; for the plane-polarized ray may be resolved into two rays circularly polarized in opposite senses, and these advance in phase by different ​amounts in passing through a given thickness of the substance: at any stage they may be recompounded into a plane-polarized ray, the azimuth of whose plane of polarization varies with the length of path traversed.
It is readily seen from this that a ray of light incident on a crystal of quartz will in general bifurcate into two refracted rays, each of which will be elliptically polarized, i.e. will be capable of resolution into two plane-polarized components which differ in phase by a definite amount. The directions of these refracted rays may be determined by Huygens' construction, provided the wave-surface is supposed to consist of a sphere and spheroid which do not touch.
The first attempt to frame a theory of naturally active bodies was made by MacCullagh in 1836. Suppose a plane wave of light to be propagated within a crystal of quartz. Let (x, y, z) denote the coordinates of a vibrating molecule, when the axis of x is taken at right angles to the plane of the wave, and the axis of z at right angles to the axis of the crystal. Using Y and Z to denote the displacements parallel to the axes of y and z respectively at any time t, MacCullagh assumed that the differential equations which determine Y and Z are
${\frac {\partial ^{2}Y}{\partial t^{2}}}=c_{1}^{2}{\frac {\partial ^{2}Y}{\partial x^{2}}}+\mu {\frac {\partial ^{3}Z}{\partial x^{3}}}$
${\frac {\partial ^{2}Z}{\partial t^{2}}}=c_{2}^{2}{\frac {\partial ^{2}Z}{\partial x^{2}}}-\mu {\frac {\partial ^{3}Y}{\partial x^{3}}}$
where μ denotes a constant on which the natural rotatory property of the crystal clepends. In order to avoid complications arising from the ordinary crystalline properties of quartz, we shall suppose that the light is propagated parallel to the optic axis, so that we can take c$_1$ equal to c$_2$.
Assuming first that the beam is circularly polarized, let it be represented by
$Y=A\sin {\frac {2\pi }{\tau }}(lx-t)$, $Z=\pm A\cos {\frac {2\pi }{\tau }}(lx-t)$,
Substituting in the above differential equations, we have
$c_{1}^{2}l^{2}\mp \mu .{\frac {2\pi }{\tau }}.l^{3}$,
or
$l={\frac {1}{c_{1}}}\pm {\frac {\pi \mu }{\tau c_{1}^{4}}}$.
Since 1/l denotes the velocity of propagation, it is evident that the reciprocals of the velocities of propagation of a right-handed and left-handed beam differ by the quantity
${\frac {2\pi \mu }{\tau c_{1}^{4}}}$;
from which it is easily shown that the angle through which the plane of polarization of a plane-polarized beam rotates in unit length of path is
${\frac {2\pi ^{2}\mu }{\tau ^{2}c_{1}^{4}}}$.
If we neglect the variation of c, with the period of the light, this expression satisfies Biot's law that the angle of rotation in unit length of path is proportional to the inverse square of the wave-length.
MacCullagh's investigation can be scarcely called a theory, for it amounts only to a reduction of the phenomena to empirical, though mathematical, laws; but it was on this foundation that later workers built the theory which is now accepted.
​The great investigators who developed the theory of light after the death of Fresnel devoted considerable attention to the optical properties of metals. Their researches in this direction must now be reviewed.
The most striking properties of metals are the power of brilliantly reflecting light at all angles of incidence, which is 80 well shown by the mirrors of reflecting telescopes, and the opacity, which causes a train of waves to be extinguished before it has proceeded many wave-lengths into a metallic medium. That these two attributes are connected appears probable from the fact that certain non-metallic bodies-e.g., aniline dyes—which strongly absorb the rays in certain parts of the spectrum, reflect those rays with almost metallic brilliance. A third quality in which metals differ from transparent bodies, and which, as we shall see, is again closely related to the other two, is in regard to the polarization of the light reflected from them. This was first noticed by Malus, and in 1830 Sir David Brewster showed that plane-polarized light incident on a metallic surface remains polarized in the same plane after reflexion if its polarization is either parallel or perpendicular to the plane of reflexion, but that in other cases the reflected light is polarized elliptically.
It was this discovery of Brewster's which suggested to the mathematicians a theory of metallic reflexion. For, as we have seen, elliptic polarization is obtained when plane-polarized light is totally reflected at the surface of a transparent body; and this analogy between the effects of total reflexion and metallic reflexion led to the surmise that the latter phenomenon might be treated in the same way as Fresnel had treated the former, namely, by introducing imaginary quantitics into the formulae of ordinary reflexion. On these principles mathematical formulae were devised by MacCullagh and Cauchy.
​To explain their method, we shall suppose the incident light to be polarized in the plane of incidence. According to Fresnel's sinc-law, the amplitude of the light (polarized in this way) reflected from a transparent body is to the amplitude of the incident light in the ratio
$J={\frac {\sin(i-r)}{\sin(i+r)}}$,
where i denotes the angle of incidence and r is determined from the equation
$\sin i=\mu \sin r$.
MacCullagh and Cauchy assumed that these equations hold good also for reflexion at a metallic surface, provided the refractive index μ is replaced by a complex quantity
$\mu =\nu (1-\kappa {\sqrt {-1}})$ say,
where ν and κ are to be regarded as two constants characteristic of the metal. We have therefore
$J={\frac {\tan i-\tan r}{\tan i+\tan r}}={\frac {(\mu ^{2}-\sin ^{2}i)^{\frac {1}{2}}-\cos i}{(\mu ^{2}-\sin ^{2}i)^{\frac {1}{2}}+\cos i}}$
If then we write
$\nu ^{2}(1-\kappa {\sqrt {-1}})^{2}-\sin ^{2}i=U^{2}e^{2\nu {\sqrt {-1}}}$,
so that equations defining U and ν are obtained by equating separately the real and the imaginary parts of this equation, we have
$J={\frac {Ue^{\nu {\sqrt {-1}}}-\cos i}{Ue^{\nu {\sqrt {-1}}}+\cos i}}$
and this may be written in the form
${\bar {J}}e^{\delta {\sqrt {-1}}}$
where
${\begin{cases}{\bar {J}}^{2}&=&{\frac {U^{2}+\cos ^{2}i-2U\cos \nu \ \cos i}{U^{2}+\cos ^{2}i+2U\cos \nu \ \cos i}}\\\tan \delta &=&{\frac {2U\cos i\ \sin \nu }{U^{2}-\cos ^{2}i}}.\end{cases}}$
$\textstyle {\bar {J}}$
$\textstyle {\bar {J}}^{2}$
The case of light polarized at right angles to the plane of incidence may be treated in the same way.
When the incidence is perpendicular, U evidently reduces to ν(1 + κ$^2$)$^{1/2}$, and ν reduces to -tan$^{-1}$κ. For silver at perpendicular incidence almost all the light is reflected, so $\textstyle {\bar {J}}^{2}$ is nearly unity: this requires cos ν to be small, and κ to be very large. The extreme case in which κ is indefinitely great but ν indefinitely small, so that the quasi-index of refraction is a pure imaginary, is generally known as the case of ideal silver.
The physical significance of the two constants ν and κ was more or less distinctly indicated by Cauchy; in fact, as the difference between metals and transparent bodies depends on the constant κ, it is evident that κ must in some way measure the opacity of the substance. This will be more clearly seen if we inquire how the elastic-solid theory of light can be extended 80 as to provide a physical basis for the formulae of MacCullagh and Cauchy. The sine-formula of Fresnel, which was the starting-point of our investigation of metallic reflexion, is a consequence of Green's elastic-solid theory: and the differences between Green's results and those which we have derived arise solely from the complex value which we have assumed for μ We have therefore to modify Green's theory in such a way as to obtain a complex value for the index of refraction.
Take the plane of incidence as plane of xy, and the metallic surface as plane of yz. If the light is polarized in the plane of incidence, so that the light-vector is parallel to the axis of z, the incident light may be taken to be a function of the argument
$ax+by+ct$,
${\frac {a}{c}}=-\left({\frac {\rho }{n}}\right)^{\frac {1}{2}}\cos i$, $\qquad {\frac {b}{c}}=-\left({\frac {\rho }{n}}\right)^{\frac {1}{2}}\sin i$;
here i denotes the angle of incidence, ρ the inertia of the aether, and n its rigidity.
Let the reflected light be a function of the argument
$a_{1}x+by+ct$,
where, in order to secure continuity at the boundary, b and c must have the same values as before. Since Green's formulae are to be still applicable, we must have
${\frac {a_{1}}{b}}=\cot r$,
where $\sin i=\mu \sin r$, but μ has now a complex value. This equation may be written in the form
$a_{1}^{2}+b^{2}={\frac {\mu ^{2}\rho c^{2}}{n}}$.
Let the complex value of μ$^2$ be written
$\mu ^{2}={\frac {\rho _{1}}{\rho }}-A{\sqrt {-1}}$,
the real part being written ρ$_1$/ρ in order to exhibit the analogy with Green's theory of transparent media: then we have
$a_{1}^{2}+b^{2}={\frac {\rho _{1}}{n}}c^{2}-{\frac {\rho c^{2}}{n}}A{\sqrt {-1}}$.
But an equation of this kind must (as in Green's theory) represent the condition to be satisfied in order that the quantity
$e^{(a_{1}x+by+ct){\sqrt {-1}}}$
may satisfy the differential equation of motion of the aether; from which we see that the equation of motion of the aether in the metallic medium is probably of the form
$\rho _{1}{\frac {\partial ^{2}e_{z}}{\partial t^{2}}}+\rho cA{\frac {\partial e_{z}}{\partial t}}=n\left({\frac {\partial ^{2}e_{z}}{\partial x^{2}}}+{\frac {\partial ^{2}e_{z}}{\partial y^{2}}}\right)$.
This equation of motion differs from that of a Greenian ​elastic solid by reason of the occurrence of the term in $\partial e_{x}/\partial t$. But this is evidently a "viscous" term, representing something like a frictional dissipation of the energy of luminous vibrations: a dissipation which, in fact, occasions the opacity of the metal. Thus the term which expresses opacity in the equation of motion of the luminiferous medium appears as the origin of the peculiarities of metallic reflexion. It is curious to notice how closely this accords with the idea of Huygens, that metals are characterized by the presence of soft particles which camp the vibrations of light.
There is, however, one great difficulty attending this explanation of metallic reflexion, which was first pointed out by Lord Rayleigh. We have seen that for ideal silver μ$^2$ is real and negative: and therefore A must be zero and ρ$_1$ negative; that is to say, the inertia of the luminiferous medium in the metal must be negative. This seems to destroy entirely the physical intelligibility of the theory as applied to the case of ideal silver.
The difficulty is a deep-seated one, and was not overcome for many years. The direction in which the true solution lies will suggest itself when we consider the resemblance which has already been noticed between metals and those substances which show "surface colour"—e.g. the aniline dyes. In the case of the latter substances, the light which is so copiously reflected from them lies within a restricted part of the spectrum; and it therefore seems probable that the phenomenon is not to be attributed to the existence of dissipative terms, but that it belongs rather to the same class of effects as dispersion, and is to be referred to the same causes. In fact, dispersion means that the value of the refractive index of a substance with respect to any kind of light depends on the period of the light; and we have only to suppose that the physical causes which operate in dispersion cause the refractive index ​to become imaginary for certain kinds of light, in order to explain satisfactorily both the surface colours of the aniline dyes and the strong reflecting powers of the metals.
Dispersion was the subject of several memoirs by the founders of the elastic-solid theory. So early as 1830 Cauchy's attention was directed to the possibility of constructing a mathematical theory of this phenomenon on the basis of Fresnel's "Hypothesis of Finite Impacts"—i.e. the assumption that the radius of action of one particle of the luminiferous medium on its neighbours is so large as to be comparable with the wave-length of light. Cauchy supposed the medium to be formed, as in Navier's theory of elastic solids, of a system of point-centres of force: the force between two of these point-centres, m at (x, y, z), and μ at (x + Δx, y + Δy,z + Δz), may be denoted by mμf(r), where r denotes the distance between m and μ. When this medium is disturbed by light-waves propagated parallel to the z-axis, the displacement being parallel to the x-axis, the equation of motion of m is evidently
${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\sum _{\mu }\mu f(r+\rho ){\frac {\Delta x+\Delta \xi }{r+\rho }}$,
where ξ denotes the displacement of m, (ξ + Δξ) the displacement of μ, and (r + ρ) the new value of r. Substituting for ρ its value, and retaining only terms of the first degree in Δξ, this equation becomes
${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\textstyle {\sum }\mu {\frac {f(r)}{r}}\Delta \xi +\textstyle {\sum }\mu {\frac {d}{dr}}\left\{{\frac {f(r)}{r}}\right\}{\frac {(\Delta x)^{2}}{r}}\Delta \xi$.
Now, by Taylor's theorem, since ξ depends only on z, wo have
$\Delta \xi ={\frac {\partial \xi }{\partial z}}\Delta z+{\frac {1}{2!}}{\frac {\partial ^{2}\xi }{\partial z^{2}}}(\Delta z)^{2}+{\frac {1}{3!}}{\frac {\partial ^{3}\xi }{\partial z^{3}}}(\Delta z)^{3}$…
Substituting, and remembering that summations which involve odd powers of Δz: must vanish when taken over all ​the point-centres within the sphere of influence of m, we obtain an equation of the form ${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\alpha {\frac {\partial ^{2}\xi }{\partial z^{2}}}+\beta {\frac {\partial ^{4}\xi }{\partial z^{4}}}+\gamma {\frac {\partial ^{6}\xi }{\partial z^{6}}}$…, where α, β, γ denote constants. Each successive term on the right-hand side of this equation involves an additional factor (Δz)/λ$^2$ as compared with the preceding term, where λ denotes the wave-length of the light: so if the radii of influence of the point-centres were indefinitely small in comparison with the wave-length of the light, the equation would reduce to ${\frac {\partial ^{2}\xi }{\partial t^{2}}}=\alpha {\frac {\partial ^{2}\xi }{\partial z^{2}}}$ which is the ordinary equation of wave-propagation in one dimension in non-dispersive media. But if the medium is so coarse-grained that λ is not large compared with the radii of influence, we must retain the higher derivates of ξ. Substituting $\xi =e^{{\frac {2\pi i}{\lambda }}(z-c_{1}t)}$ in the differential equation with these higher derivates retained, we have $c_{1}^{2}=\alpha -\beta \left({\frac {2\pi }{\lambda }}\right)^{2}+\gamma \left({\frac {2\pi }{\lambda }}\right)^{4}$…, which shows that c$_1$, the velocity of the light in the medium, depends on the wave-length λ; as it should do in order to explain dispersion.
Dispersion is, then, according to the view of Fresnel and Cauchy, a consequence of the coarse-grainedness of the medium. Since the luminiferous medium was found to be dispersive only within material bodies, it seemed natural to suppose that in these bodies the aether is loaded by the molecules of matter, and that dispersion depends essentially on the ratio of the wave-length to the distance between adjacent material molecules. ​This theory, in one modification or another, held its ground until forty years later it was overthrown by the facts of anomalous dispersion.
The distinction between aether and ponderable matter was more definitely drawn in memoirs which were published independently in 1841–2 by F. E. Neumann and Matthew O'Brien. These authors supposed the ponderable particles to remain sensibly at rest while the aether surges round them, and is acted on by them with forces which are proportional to its displacement. Thus the equation of motion of the aether becomes
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {4}{3}}n){\text{grad div }}\mathbf {e} -n{\text{ curl curl }}\mathbf {e} -C\mathbf {e}$,
where C denotes a constant on which the phenomena of dispersion depend. For polarized plane waves propagated parallel to the axis of x, this equation becomes
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=n{\frac {\partial ^{2}\mathbf {e} }{\partial x^{2}}}-C\mathbf {e}$;
and substituting
$\mathbf {e} =e^{{\frac {2\pi {\sqrt {-1}}}{\tau }}(t-{\frac {x}{V}})}$,
where τ denotes the period and V the velocity of the light, we have
${\frac {n}{V^{2}}}=\rho -{\frac {C}{r\pi ^{2}}}r^{2}$,
an equation which expresses the dependence of the velocity on the period.
The attempt to represent the properties of the aether by those of an elastic solid lost some of its interest after the rise of the electromagnetic theory of light. But in 1867, ​before the electromagnetic hypothesis had attracted much attention, an elastic-solid theory in many respects preferable to its predecessors was presented to the French Academy by Joseph Boussinesq (b. 1842). Until this time, as we have seen, investigators had been divided into two parties, according as they attributed the optical properties of different bodies to variations in the inertia of the luminiferous medium, or to variations in its elastic properties. Boussinesq, taking up a position apart from both these schools, assumed that the aether is exactly the same in all material bodies as in interplanetary space, in regard both to inertia and to rigidity, and that the optical properties of matter are due to interaction between the aether and the material particles, as had been imagined more or less by Neumann and O'Brien. These material particles he supposed to be disseminated in the aether, in much the same way as dust-particles floating in the air.
If e denote the displacement at the point (x, y, z) in the aether, and e′ the displacement of the ponderable particles at the same place, the equation of motion of the aether is
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {1}{3}}n){\text{ grad div }}\mathbf {e} +n\nabla ^{2}\mathbf {e} -\rho _{1}{\frac {\partial ^{2}\mathbf {e} ^{\prime }}{\partial t^{2}}},\qquad$(1)
where ρ and ρ$_1$ denote the densities of the aether and matter respectively, and k and n denote as usual the elastic constants of the aether. This differs from the ordinary Cauchy-Green equation only in presence of the term ρ$_1$&part$^2$e′/∂t$^2$, which represents the effect of the inertia of the matter. To this equation we must adjoin another expressing the connexion between the displacements of the matter and of the aether: if we assume that these are simply proportional to each other—say,
$\mathbf {e} ^{\prime }=A\mathbf {e} ,\qquad \qquad$(2)
$(\rho +\rho _{1}A){\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {1}{3}}n){\text{ grad div }}\mathbf {e} +n\nabla ^{2}\mathbf {e}$,
which is essentially the same equation as is obtained in those. older theories which suppose the inertia of the luminiferous. medium to vary from one medium to another. So far there would seem to be nothing very new in Boussinesq's work. But when we proceed to consider crystal-optics, dispersion, and rotatory polarization, the advantage of his method becomes evident: he retains equation (1) as a formula universally true—at any rate for bodies at rest—while equation (2) is varied to suit the circumstances of the case. Thus dispersion can be explained if, instead of equation (2), we take the relation
$\mathbf {e} ^{\prime }=A\mathbf {e} -D\nabla ^{2}\mathbf {e}$,
where D is a constant which measures the dispersive power of the substance: the rotation of the plane of polarization of sugar solutions can be explained if we suppose that in these bodies equation (2) is replaced by
$\mathbf {e} ^{\prime }=A\mathbf {e} +B{\text{ curl }}\mathbf {e}$,
where B is a constant which measures the rotatory power, and the optical properties of crystals can be explained if we suppose that for them equation (2) is to be replaced by the equations
$e_{x}^{\prime }=A_{1}e_{x}$, $\qquad e_{y}^{\prime }=A_{2}e_{y}$, $\qquad e_{z}^{\prime }=A_{3}e_{z}$
When these values for the components of e′ are substituted in equation (1), we evidently obtain the same formulae as were derived from the Stokes-Rankine-Rayleigh hypothesis of inertia different in different directions in a crystal; to which Boussinesq's theory of crystal-optics is practically equivalent.
The optical properties of bodies in motion may be accounted for by modifying equation (1), so that it takes the form
$\rho {\frac {\partial ^{2}\mathbf {e} }{\partial t^{2}}}=-(k+{\tfrac {1}{3}}n){\text{ grad div }}\mathbf {e} +n\nabla ^{2}\mathbf {e} -\rho _{1}\left({\frac {\partial }{\partial t}}+w_{x}{\frac {\partial }{\partial x}}+w_{y}{\frac {\partial }{\partial y}}+w_{z}{\frac {\partial }{\partial z}}\right)^{2}\mathbf {e} ^{\prime }$,
$\rho {\frac {\partial ^{2}e_{x}}{\partial t^{2}}}=n{\frac {\partial ^{2}e_{x}}{\partial z^{2}}}-\rho _{1}A\left({\frac {\partial }{\partial t}}+w{\frac {\partial }{\partial z}}\right)^{2}e_{x}$;
substituting
$e_{x}=f(z-Vt)$,
where V denotes the velocity of propagation of light in the medium estimated with reference to the fixed aether, we obtain for V the value
$\left({\frac {n}{\rho +\rho _{1}A}}\right)^{\frac {1}{2}}+{\frac {\rho _{1}A}{\rho +\rho _{1}A}}w$.
The absolute velocity of light is therefore increased by the amount ρ$_1$Aw/(ρ + ρ$_1$A) owing to the motion of the medium; and this may be written (μ$^2$ - 1) w/μ$^2$, where μ denotes the refractive index; so that Boussinesq's theory leads to the same formula as had been given half a century previously by Fresnel.
It is Boussinesq's merit to have clearly asserted that all space, both within and without ponderable bodies, is occupied by one identical aether, the same everywhere both in inertia and elasticity; and that all aethereal processes are to be represented by two kinds of equations, of which one kind expresses the invariable equations of motion of the aether, while the other kind expresses the interaction between aether and matter. Many years afterwards these ideas were revived in connexion with the electromagnetic theory, in the modern forms of which they are indeed of fundamental importance.
* ↑ Trans. Camb. Phil. Soc., viii, p. 287 (1845).
* ↑ Mém. de l'Acad. vii, p. 375. The memoir was presented in 1821, and published in 1827.
* ↑ Hamilton's opinion, written in 1833, is worth repeating: "The principal theories of algebraical analysis (under which I include Calculi) require to be entirely remodelled; and Cauchy has done much already for this great object. Poisson also has done much; but he does not seen to me to have nearly so logical a mind as Cauchy, great as his talents and clearness are; and both are in my judgment very far inferior to Fourier, whom I place at the head of the French School of Mathematical Philosophy, eren above Lagrange and Laplace, though I rank their talents above those of Cauchy and Poisson." (Life of Sir W. R. Hamilton, ii, p. 58.)
* ↑ Cauchy, Exercices de Mathématiques iii, p. 160 (1828).
* ↑ This notation was introduced at a later period, but is used here in order to avoid subsequent changes.
* ↑ In order to construct a body whose elastic properties are not limited by this equation, William John Macquorn Rankine (b.1820, d. 1872) considered a continuous fluid in which a number of point-centres of force are situated: the fluid is supposed to be partially condensed round these centres, the elastic atmosphere of each nucleus being retained round it by attraction. An additional volume-elasticity due to the fluid is thus acquired; and no relation between k and n is now necessary. Cf. Rankine's Miscellaneous Scientific Papers, pp. 81 994. Sir William Thomson (Lord Kelvin), in 1889, formed a solid not obeying Navier's condition by using pairs of dissimilar atoms. Of. Thomson's Papers, iii, p. 396. Cf. also Baltimore Lectures, pp. 123 sqq.
* ↑ It may, however, be objected that india-rubber and other bodies which fail to fulfil Navier's relation are not true solids. On this historic controversy, cf. Todhunter and Pearson's History of Elasticity, i, p. 496.
* ↑ Mém. de l'Acad., viii (1828), p. 623. Poisson takes the equation in the restricted form given by Navier; but this does not affect the question of wave-propagation.
* ↑ Cf. Stokes, "On the Dynamical Problem of Diffraction," Camb. Phil. Trans., ix (1849).
* ↑ It may easily be shown that any disturbance, in either isotropic or crystalline media, for which the direction of vibration of the molecules lies in the wave-front or surface of constant phase, must satisfy the equation $\mathrm {div} \ \mathbf {e} =0$ where e denotes the displacement; if, on the other hand, the direction of vibration of the molecules is perpendicular to the wave-front, the disturbance must satisfy the equation $\mathrm {curl} \ \mathbf {e} =0$ These results were proved by M. O'Brien, Trans. Camb. Phil. Soc., 1842.
* ↑ Exercices de Math., iii (1828), p. 188.
* ↑ These are substantially equations (68) on page 208 of the third volume of the Exercices.
* ↑ G, H, I are tensions when they are positive, and pressures when they are negative.
* ↑ Mém. de l'Acad., x, p. 293. In the previous year (Mém. de l'Acad., ix, p. 114) Cauchy had stated that the equations of elasticity lead in the case of uniaxal crystals to a wave-surface of which two sheets are a sphere and spheroid as in Huygens' theory.
* ↑ The equations and results of Cauchy's First Theory of crystal-optics were independently obtained shortly afterwards by Franz Ernst Neumann (b. 1798, d. 1895): cf. Ann. d. Phys. xxv (1832), p. 418, reprinted as No. 76 of Ostwald's Klassiker der exakten Wissenschaften, with notes by A. Wangerin.
* ↑ Comptes Readus, ii (1836), p. 341: Mém. de l'Acad. xvii (1839), p. 163.
* ↑ Mém. de l'Acad. xvii, p. 161.
* ↑ Bull. des Sciences Math, xiv. (1830), p. 6.
* ↑ As will appear, his views on this subject subsequently changed.
* ↑ Comptes Rendus, ii. (1836), p. 341.
* ↑ Comptes Rendus, ii. (1836), p. 341: "Mémoire sur la dispersion de la lumière" (Nouveaux exercices de Math., 1836), p. 203.
* ↑ These boundary-conditions of Cauchy's are, as a matter of fact, satisfied by the electric force in the electro-magnetic theory of light. The continuity of curl e is equivalent to the continuity of the magnetic vector across the interface, and the continuity of $\partial e_{x}/\partial x$ leads to the same equation as the continuity of the component of electric force in the direction of the intersection of the interface with the plane of incidence.
* ↑ One yet remains to be mentioned.
* ↑ Tho outlines of the theory were published by MacCullagh in Brit. Assoc. Rep. 1835; and his results were given in Phil. Mag. (Jan., 1837), and in Proc. Royal Irish Acad. xviii. (Jan., 1837). Neumann's memoir was presented to the Berlin Academy towards the end of 1835, and published in 1837 in Abh. Berl. Ak, aus dem Jahre 1835, Math. Klasse, p. 1. So far as publication is concerned, the priority would seem to belong to MacCullagh; but there are reasons for believing that the priority of discovery really rests with Neumann, who bad arrived at his equations a year before they were communicated to the Berlin Academy.
* ↑ Trans. Camb. Phil. Soc., 1838; Green's Math. Papers, p. 245.
* ↑ These first three conditions are of course not dynamical but geometrical.
* ↑ Trans. Roy. Irish Acad. xxi.: MacCullagh's Coll. Works, p. 145.
* ↑ MacCullagh's equations may readily be interpreted in the electro-magnetic theory of light: e corresponds to the magnetic force, μ curl e to the electric force, and curl e to the electric displacement.
* ↑ Comptes Rendus, Sept. 16, 1889: Kelvin's Math. and Phys. Papers, iii, p. 466.
* ↑ Proc. Roy, Soc. Edinb., Mar. 17, 1890: Kelvin's Math. and Phys. Papers, iii, p. 468.
* ↑ Comptes Rendus, ix, p. 676 (25 Nov., 1839), and p. 726 (2 Dec., 1839).
* ↑ Green's Math. Papers, p. 246.
* ↑ Comptes Rendus, x, p. 347 (March 2, 1840): xxvii, p. 621 (1848); xxviii,p. 25 (1849). Mém. de l'Acad., xxii (1848), pp. 17, 29.
* ↑ Labile or neutral is a term used of such equilibrium as that of a rigid body on a perfectly smooth horizontal plane.
* ↑ Phil. Mag. xxvi (1888), p. 414.
* ↑ This condition is in any case necessary for stability, as was shown by R. T. Glazebrook: ef. Thomson, Phil. Mag. xxvi, p. 500.
* ↑ The labile-aether theory of light may be compared with the electro-magnetic theory, by interpreting the displacement e as the electric force, and ρe as the electric displacement.
* ↑ Baltimore Lectures (edition 1904), p. 214.
* ↑ Ibid. (ed. 1904), p. 411.
* ↑ Michell and Boscovich in the eighteenth century had taught the doctrine of the mutual penetration of matter, i.e. that two substances may be in the same place at the same time without excluding each other: cf. Priestley's History i., P. 392.
* ↑ Cf. Baltimore Lectures (ed. 1904), pp. 413-14, 463, and Appendices A and E.
* ↑ Cambridge Pbil. Trans., 1839; Green's Math. Papers, p. 293.
* ↑ For there are 21 terms in a homogeneous function of the second degree in six variables.
* ↑ As Green showed, the hypothesis of transversality really involves the existence of planes of symmetry, so that it alone is capable of giving 14 relations between the 21 constants: and 3 of the remaining 7 constants may be removed by change of axes, leaving only four.
* ↑ It was afterwards shown by Barré de Saint-Venant (b. 1797, d. 1886), Journal de Math., vii (1883), p. 399, that if the initial stresses be supposed to vanish, the conditions which must be satisfied among the remaining nine constants a, b, c, f, g, h, f′, g′, h′, in order that the wave-surface may be Fresnel's, are the following:— $\scriptstyle {\begin{cases}(3b-f)(3c-f)=(f+f^{\prime })^{2}\\(3c-g)(3a-g)=(g+g^{\prime })^{2}\\(3a-h)(3b-h)=(h+h^{\prime })^{2}\\(3a-g)(3b-h)(3c-f)+(3a-h)(3b-f)(3c-g)=2(f+f^{\prime })(g+g^{\prime })(h+h^{\prime }).\end{cases}}$ These reduce to Green's relations when the additional equation b = c is assumed. Saint-Venunt disputed the validity of Green's relations, a9serting that they are compatible only with isotropy. On this controversy cf. R. T. Glazebrook, Brit. Assoc. Report, 1885, p. 171, and Karl Pearson in Todhunter and Pearson's History of Elasticity, ii, § 147.
* ↑ Proc. R. S. Edin. xv (1887), p. 21: Phil. Mag. xxv (1888) p. 116: Baltimore Lectures (ed. 1904), pp. 228-259.
* ↑ Baltimore Lectures (ed. 1904), p. 253.
* ↑ Baltimore Lectures (ed. 1904), p. 258.
* ↑ It was in the year Thomson took his degree (1845) that he bought, and read with delight, the electrical memoir which Green had published at Nottingham in 1828.
* ↑ Trans. Camb. Phil. Soc., ix (1849), p. 1. Stokes's Math. and Phys. Papers, ii, p. 243.
* ↑ Ann, d. Phys. cxi (1860), p. 315. Phil. Mag. xxi (1861), p. 321.
* ↑ Phil. Trans., 1852, p. 463. Stokes's Math. and Phys. Papers, iii, p. 267. Cf. the foot-note added on p. 361 oi the Math. and Phys. Papers.
* ↑ The theory of polarization by small particles was afterwards investigated by Lord Rayleigh, Phil. Mag. xli (1871).
* ↑ In Fresnel's theory of crystal-optics, in which the aether-vibrations are at right angles to the plane of polarization, the velocity of propagation depends only on the direction of vibration, not on the plane through this and the direction of transmission.
* ↑ Stokes, in a letter to Lord Rayleigh, inserted in his Memoir and Scientific Correspondence, ii, p. 99, explains that the idea presented itself to him while he was writing the paper on Fluid Motion which appeared in Trans. Camb. Phil. Soc., viii (1843), p. 105. He suggested the wave-surface to which this theory leads in Brit. Assoc, Rep., 1862, p. 269.
* ↑ Phil. Mag. (4), i (1851), p. 441.
* ↑ Phil. Mag. (4), xli (1871), p. 519.
* ↑ Proc. R. S., June, 1872. After these experiments Stokes gave it as his opinion (Phil. Mag. xli (1871), p. 521) that the true theory of crystal-optics was yet to be found. On the accuracy of Fresnel's construction of. Glazebrook, Phil. Trans. clxxi (1879) p. 421, and Hastings, Am. Journ. Sci. (3) xxxv (1887) p. 60.
* ↑ Phil. Mag. xxvi (1888), p. 521; xxvii (1889), p. 110.
* ↑ This theory of crystal-optics may be assimilated to the electro-magnetic theory by interpreting the elastic displacement e as electric force, and the vector (ρ$_1$e$_x$, ρ$_2$e$_y$, ρ$_3$e$_z$) as electric displacement.
* ↑ Mém. de l'Institut, 1812, Parti, p. 116, sqq.
* ↑ Mém. de l'Institut, 1812, Part I, p. 218, 899.; Annales de Chin., ix (1818), p. 372; < (1819), p. 63.
* ↑ Camb. Phil. Soc. Trans. i, p. 43.
* ↑ Mém. de l'Inst. vii, p. 73.
* ↑ Baltimore Lectures (ed. 1904), p. 31.
* ↑ The term rotatory may be applied with propriety to the property discovered by Faraday, which will be discussed later.
* ↑ Annales de Chim, xxviii (1825), p. 147.
* ↑ Trans, Royal Irish Acad., xvii.; MacCullagh's Coll. Works, p. 63.
* ↑ The later developments of this theory will be discussed in a subsequent chapter; but mention may here be made of an attempt which was made in 1856 by Carl Neumann, then a very young man, to provide a rational basis for MacCullagh's equations. Neumann showed that the equations may be derived from the hypothesis that the relative displacement of one aethereal particle with respect to another acts on the latter according to the same law as an element of an electric current acts on a magnetic pole. Cf. the preface to C. Noumann's Die magnetische Drehung der Polarisationsebene des Lichtes, Halle, 1863.
* ↑ Phil. Trans., 1830.
* ↑ Proc. Roy. Irish Acad., i (1936), p. 2; ii (1843), p. 376: Trans. Roy. Irish Acad., xviji (1837), p. 71: MacCullagh's Coll. Works, PP. 58, 132, 230.
* ↑ Comptes Rendus, vii (1838), p. 953; viii (1839), pp. 553, 658, 961; xxvi (1848), p. 86.
* ↑ This was done by Lord Rayleigh, Phil. Mag. xliii (1872), p. 321.
* ↑ It is easily seen that the amplitude is reduced by the factor e$^{-2πk}$ when light travels one wave-length in the metal: K is generally called the coefficient of absorption.
* ↑ Loc. cit.
* ↑ Bull. des Sc. Math. xiv (1830), p. 9: "Sur la dispersion de la lumière," Nouv. Erercices de Math., 1836.
* ↑ Cf. p. 132.
* ↑ Berlin Abhandlungen aus dem Jahre 1841, Zweiter Teil, p. 1: Berlin, 1843.
* ↑ Trans. Camb. Phil. Soc. vii (1842), p. 397.
* ↑ O'Brien, loc. cit., §§ 15, 28.
* ↑ Journal de Math. (2) xiii (1868), pp. 313, 425: cf. also Comptes Rendus, cxvii (1893), pp. 80, 139, 193. Equations kindred to some of those of Boussinesq were afterwards deduced by Karl Pearson, Proc. Lond. Math. Soc, xx (1889), p. 297, from the hypotbesis that the strain-energy involves the velocities.
* ↑ Cf. p. 115 sqq.
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