A History of the Theories of Aether and Electricity (1910) — part 10 of 29

• Hamilton's opinion, written in 1833, is worth repeating : " The principal theories of algebraical analysis (under which I include Calculi) require to he entirely remodelled ; and Cauchy has done much already for this great object. Poisson also has done much ; but he does not seem 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, even above Lagrange and Laplace, though I rank their talents above those of Cauchy and Poisson." (Life of Sir W. It. Hamilton, ii, p. 58.)

t Cauchy, Exercices de Mathematiques iii, p. 160 (1828).

J This notation was introduced at a later period, but is used here in order to avoid subsequent changes.

140 The Aether as an Elastic Solid.

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*

Actual bodies do not necessarily obey this condition; e.g.

for india-rubber, k is much larger than - n ;f and there seems to

o

be no reason why we should impose it on the aether.

In the same year PoissonJ succeeded in solving the diffe- rential 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

• 5 Vb = °'

o Tlj

• 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 con- tinuous 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 sqq.

Sir "William Thomson (Lord Kelvin), in 1889, formed a solid not obeying Navier's condition by using pairs of dissimilar atoms. Cf. Thomson's Papers, iii, p. 395. Cf. also Baltimore Lectures, pp. 123 sqq.

t 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.

J Mem. de 1'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.

The Aether as an Elastic Solid. 141

The terms which involve b and those which involve c must be separately zero, since they represent respectively the irrota- tional and the circuital parts of the equation. Thus, c satisfies. the pair of equations

02-

p T-J- = ?iV2c, div c = 0 ;

vt

while b is to be determined from

dt A particular solution of the equations for c is easily seen to be

cx = A sin A (2 - t /-),

/-), cy = B sinXfz - t /-), cz = 0, \PJ V \PJ

which represents a transverse plane wave propagated with velocity ^/(n/p). 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

-t E

V

p which represents a longitudinal wave propagated with velocity

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/p)b ; while those in b are longitudinal, and are propagated with velocity {(k+ $n)/p}%. 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

• Cf . Stokes, "On the Dynamical Problem of Diffraction," Camb. Phil. Trans., ix (1849).

142 The Aether as an Elastic Solid.

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 1828f 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 these 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*

dx\ 2x ty 9 dz and two similar equations. The three constants G, H, I re- present the stresses across planes parallel to the coordinate planes in the undisturbed state of the aether. §

• 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

div 6 = 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

curl e = 0. These results were proved by M. O'Brien, Trans. Camb. Phil. Soc., 1842.

t Exercices de Math., iii (1828), p. 188.

J 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.

The Aether as an Elastic Solid. 1 43

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)% if the vibration takes place parallel to the axis of y, and (g+ G)$ 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)% if the vibration is parallel to the axis of x, and (/+ H)^ if it is parallel fo the axis of z\ and when the wave-front is parallel to the plane of xy, the velocity must be (g + /)* if the vibration is parallel to the axis of x, and (/ + /)* 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

/+#=/+/, ff + I = g+G, k + H=h+G; or, G = H=L

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, /, being all equal, are taken to be zero.

Tf, 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

• Mem. de 1'Acad., x, p. 293.

In the previous year (Mem. de 1'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.

f The equations and results of Cauchy's First Theory of crystal-optics were independently obtained shortly afterwards hy 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.

144 The Aether as an Elastic Solid.

the theory based on this supposition is known as Caucliy'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

3£c = f(b + c +/) ; 3ca = g(c + a + g) ; Sab = h(a + b + Ji).

Thus in the first theory we have these together with the

equations

£ = 0, H=Q, 1=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-Q-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 placef 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 subse- quently to compare the properties which are assigned to the aether in crystal- op tics with those which must be postulated in order to account for reflexion and refraction.

• Comptes Rendus, ii (1836), p. 341 : Mem. de 1'Acad. xviii (1839), p. 153. f Mem. de 1'Acad. xviii, p. 161.

The Aether as an Elastic Solid. 145

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 papersf followed this lead more or less closely, assuming that the density p 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 inter- section 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~ = /( - x cos i -y sin i + rL t \

where i denotes the angle of incidence ; the reflected wave may be represented by

ez = FX cos i - y sin i + t V /

and the refracted wave by

ez = fi I — x cos r — y sin r + KLt\

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 ^/-components of the stress across the #y-plane are equal in

• Bull, des Sciences Math. xiv. (1830), p. 6.

t As will appear, his views on this subject subsequently changed.

L

146 The Aether as an Elastic Solid.

the media on either side the interface. This implies in the present case that the quantities

tie* dez

n — and n —

dx ty

are to be continuous across the interface : so we have

n cos i'. (/' - 1") = n' cos r . /', ; n sin i.(f' + F) = n' sin r . f.

Eliminating /'„ we have

F' _ sin (r - i) f 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 to 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 ex and the normal pressure across the interface have equal values in the two contiguous media.

The theory thus advanced was encumbered with many diffi- culties. 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

*Comptes Rendus, ii. (1836), p. 341.

The Aether as an Elastic Solid. 147

reject the first theory of crystal-optics in favour of the second. After 1836 he consistently adhered to the view that the vibra- tions 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) dex/dx 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. f 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 appear- ance 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

• Comptes Rendus, ii. (1836), p. 341 : " Meraoire sur la dispersion delalumiere " (Nouveaux exercices de Math., 1836), p. 203.

t 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 <;url e is equivalent to the continuity of the magnetic vector across the interface, and the continuity of (tex/dx 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.

L 2

] 48 The Aether as an Elastic Solid.

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 con- sidered the question from the purely analytical point of view. Given certain differential equations, what supplementary con- ditions 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 mathe- matician 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 timef almost simultaneously by James MacCullagh (6. 1809, d. 1847), of Trinity College, Dublin, and Franz Neumann (b. 1798, d. 1895), of Konigsberg. 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 ease of isotropic media, to which our attention will for the present be confined.

• One yet remains to be mentioned.

f The outlines of the theory were published by MacCullagh in Brit. Assoc. Rep. 1835 ; and his results were given in Phil. Mag. x (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 had arrived at his equations a year before they were communicated to the Berlin Academy.

The Aether as an Elastic Solid. 149

MacCullagh and Neumann felt that the great objection to FresnePs 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 con- dition 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 Presnel; 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

150 The Aether as an Elastic Solid.

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

4 \tex dey

• (- + ^} -4r-*-4~~-4

where e denotes the displacement, and k and n denote the two

• Trans. Camb. Phil. Soc., 1838 ; Green's Math. Papers, p. 245.

The Aether as an, Elastic Solid. 151

elastic constants already introduced; by substituting this value of <f> in the general variational equation

III'0 \w &t + 1* ** + TF*-| ****** = -

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

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

or,

/ 4 \

pe = - ( k + - n ) grad div e - n curl curl e ; \ • /

pe = -lk + -n\ grad div e + nVze,

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

, 2 \ dex fiez 3ex\ fdev dey

--TI dive+ 27i — , w(-Ji + — ), and n (^ + -£

3 ) dx \to fa J \ty dx

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, pro- vided the rigidity n is assumed to be the same for all media, but the inertia p 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

• These first three conditions are of course not dynamical but geometrical.

152 The Aether as an Elastic Solid.

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 inci- dence, so that the motion of the aethereal particles is parallel to the intersection of the plane of incidence with the front of the wave. In this case it is impossible to satisfy all the six boundary-conditions without assuming that longitudinal vibra- tions 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

6 = A + lz

where, if i denote the angle of incidence, we have

I = . /— cos it m = - /— sin i. \n Mn

There will be a transverse reflected wave,

and a transverse refracted wave,

y); ez = - C — f(t + 1& + my),

where, since the velocity of transverse waves in the second medium is v/W/oz, we can determine ^ from the equation

^•f^.&j

n

there will also be a longitudinal reflected wave,

8 9

ey = D -f(t -\z + my); ez = D -f(t - \z + my),

The Aeiher as an Elastic Solid. 153

where A is determined by the equation

and a longitudinal refracted wave,

7\ 7\

ey = JE - /(* + Aiz + my) ; ez = E - f(t where AI is determined by

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

A- E _ ljj>i m? (pi - p2)2 A + B Ip2 I pz (\pz + A!/?I)

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 wavev-G^een avoided this difficulty by adopting Fresnel's suggestion that the resistance of the aether to compression may V\ 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\ and &2 are very large, the •quantities A and A: have the values m </ - 1, and we have

A- B li pi m (PI - p2f

A + B I 2

154 The Aether as an Elastic Solid.

Thus if BjA denote the modulus of £/A, we have

p\

if>l

This expression represents the ratio of the intensity of the transverse reflected wave to that of 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 elastic-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 Eoyal Irish Academy in 1839, is the intro- duction 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

• Trans. Roy. Irish Acad. xxi. : MacCullagh's Coll. Works, p. 145.

The Aether as an Elastic Solid. 155

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 <f> must be formed from the only invariant which depends solely on the rotation and is quadratic in the derivates, that is from (curl e)2 ; thus we may write

to,

*~

The equation of motion is now to be determined, as in the case of Green's aether, from the variational equation

the result is

p—z = - fi curl curl 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

156 The Aether as an Elastic Solid.

which shows that transverse waves are propagated with velocity

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 p 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

p a"? = ~3^ (» curl e)* + l°* curl e)" / ' . |

and as the quantities p, (n curl e)2, and (/n curl e)y are continuous across the interface, the continuity of c>2ex/dtz follows. Thus the only independent boundary-conditions in MacCullagh's theory are the continuity of the tangential components of e and of fj 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

\dz dx J \c>x ty J

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

• MacCullagh's equations may readily be interpreted in the electro -magnetic theory of light : e corresponds to the magnetic force, p curl e to the electric force, and curl e to the electric displacement.

The Aether as an Elastic Solid. 157

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.