of symmetry to which the medium is referred. For waves perpendicular to the principal axes, the directions of vibration and squared velocities of propagation are as follows:

Green assumes, in accordance with Fresnel's theory, and with observation if the vibrations in polarized light are supposed perpendicular to the plane of polarization, that for waves perpendicular to any two of the principal axes, and propagated by vibrations in the direction of the third axis, the velocity of propagation is the same. This gives three, equivalent to two, relations among the constants, namely,

A-L=B-M-C-N=v suppose,

(9)

which are equivalent to Cauchy's equations (3). The conditions that the vibrations are strictly transversal and normal respectively do not involve the six constants expressing the pressures in equilibrium, and therefore remain the same as before, namely (7). Adopting the relations (7) and (9), Green proves that for the two transversal waves the velocities of propagation and the azimuths of the planes of polarization are precisely those given by the theory of Fresnel, the vibrations in polarized light being now supposed perpendicular to the plane of polarization.

As to the wave propagated by normal vibrations, the square of its velocity of propagation is easily shown to be equal to

μ+Al2+Bm2 + Cn2;

and as the constant μ does not enter into the expression for the velocity of propagation of transversal vibrations, the same supposition as before, namely that the medium is rigorously or sensibly incompressible, removes all difficulty arising from the absence of any observed phenomenon answering to this wave.

The existence of planes of symmetry is here in part assumed. I say in part, because Green shows that the six constants, expressing the pressures in equilibrium, enter the equation of the ellipsoid of elasticity under the form K(x2+y2+z2), where K is a homogeneous function of the six constants of the first order, and involves likewise the cosines l, m, n. Hence the directions of vibration are the same as when the six constants vanish; the velocities of propagation alone are changed; and as the existence of planes of symmetry for the case in which the six constants vanish was demonstrated, it is only requisite to make the very natural supposition that the planes of symmetry which must exist as regards the directions of vibration, are also planes of symmetry as regards the pressure in equilibrium.

We see then that this theory, which may be called Green's second theory, is in most respects as satisfactory (assuming for the present that Fresnel's construction does represent the laws of double refraction) as the former. I say in most respects, because, although the theory is perfectly rigorous, like the former, the equations (9) are of the nature of forced relations between the constants, not expressing anything which could have been foreseen, or

even conveying when pointed out the expression of any simple physical relation.

The year 1839 was fertile in theories of double refraction, and on the 9th of December Prof. MacCullagh presented his theory to the Royal Irish Academy. It is contained in "An Essay towards a Dynamical Theory of Crystalline Reflexion and Refraction "*. As indicated by the title, the determination of the intensities of the light reflected and refracted at the surface of a crystal is what the author had chiefly in view, but his previous researches had led him to observe that this determination was intimately connected with the laws of double refraction, and to seek to link together these laws as parts of the same system. He was led to apply to the problem the general equation of dynamics under the form (5), to seek to determine the form of the function 4 (V in his notation), and then to form the partial differential equations of motion, and the conditions to be satisfied at the boundaries of the medium, by the method of Lagrange. He does not appear to have been aware at the time that this method had previously been adopted by Green. Like his predecessors, he treats the ether within a crystallized body as a single medium unequally elastic in different directions, thus ignoring any direct influence of the ponderable molecules in the vibrations. He assumes that the density of the ether is a constant quantity, that is, both unchanged during vibration, and the same within all bodies as in free space. We are not concerned with the latter of these suppositions in deducing the laws of internal vibrations, but only in investigating those which regulate the intensity of reflected and refracted light. He assumes further that the vibrations in plane waves, propagated within a crystal, are rectilinear, and that while the plane of the wave moves parallel to itself the vibrations continue parallel to a fixed right line, the direction of this right line and the direction of a normal to the wave being functions of each other, a supposition which doubtless applies to all crystals except quartz, and those which possess a similar property.

In this method everything depends on the correct determination of the form of the function V. From the assumption that the density of the ether is unchanged by vibration, it is readily shown that the vibrations are entirely transversal. Imagine a system of plane waves, in which the vibrations are parallel to a fixed line in the plane of a wave, to be propagated in the crystal, and refer the crystal for a moment to the rectangular axes of x', y', z', the plane of x'y' being parallel to the planes of the waves, and the axis of y' to dn' the direction of vibration; and let be the angle whose tangent is Κ With dz respect to the form of V, MacCullagh reasons thus :-"The function V can only depend upon the directions of the axes of a', y', z' with respect to fixed lines in the crystal, and upon the angle which measures the change of form produced in the parallelepiped by vibration. This is the most general supposition which can be made concerning it. Since, however, by our second supposition, any one of these directions, suppose that of x', determines the other two, we may regard V as depending on the angle and the direction of the axis of a alone," from whence he shows that V must be a function of the quantities X, Y, Z, defined by the equations

dn d

x=dz dy

dz dž, z=
αξ de dn

2, Y=da dz

dy

dx

This reasoning, which is somewhat obscure, seems to me to involve a fallacy.

* Memoirs of the Royal Irish Academy, vol. xxi. p. 17.

If the form of V were known, the rectilinearity of vibration and the constancy in the direction of vibration for a system of plane waves travelling in any given direction would follow as a result of the solution of the problem. But in using equation (5) we are not at liberty to substitute for V (or 4) an expression which represents that function only on the condition that the motion be what it actually is, for we have occasion to take the variation &V of V, and this variation must be the most general that is geometrically possible though it be dynamically impossible. That the form of V, arrived at by MacCullagh, is inadmissible, is, I conceive, proved by its incompatibility with the form deduced by Green from the very same supposition of the perfect transversality of the transversal vibrations; for Green's reasoning is perfectly straightforward and irreproachable. Besides, MacCullagh's form leads to consequences absolutely at variance with dynamical principles*.

But waiving for the present the objection to the conclusion that V is a function of the quantities X, Y, Z, let us follow the consequences of the theory. The disturbance being supposed small, the quantities X, Y, Z will also be small, and V may be expanded in a series according to powers of these quantities; and, as before, we need only proceed to the second order if we regard the disturbance as indefinitely small. The first term, being merely a constant, may be omitted. The terms of the first order MacCullagh concludes must vanish. This, however, it must be observed, is only true on the supposition that the medium in its undisturbed state is free from pressure. The terms of the second order are six in number, involving squares and products of X, Y, Z. The terms involving YZ, ZX, XY may be got rid of by a transformation of coordinates, when V will be reduced to the form

the constant term being omitted, and the arbitrary constants being denoted by -a2, -b2, c2. Thus on this theory the existence of principal axes is proved, not assumed. If MacCullagh's expression for V (10) be compared with Green's expression for (8) for the case of no pressure in equilibrium, so that A=0, B=0, C=0, it will be seen that the two will become identical, provided du dv dw\2 first we omit the term μ + + in Green's expression, and secondly, dx dy dz

Φ

we treat the symbols of differentiation as literal coefficients, so as to confound, dv dw dv dw for instance, The term involving μ does not appear in the

dy dz

and

dz dy

du dv dw expressions for transversal vibrations, since for these + + =0, and dx dy dz

therefore does not affect the laws of the propagation of such vibrations, although it would appear in the problem of calculating the intensity of reflected and refracted light; and be that as it may, it follows from Green's rule for forming the equation of the ellipsoid of elasticity, that the laws of the propagation of transversal vibrations will be precisely the same whether we adopt his form of or V (for the case of no pressure in equilibrium) or MacCullagh's. Indeed, if we omit the term μdx dy dz (du, dv, dw)2 + d), the partial differential equations of motion, on which alone depend the laws of internal propagation, would be just the same as the two theoriest. Accordingly MacCullagh obtained, though * See Appendix.

+ See Appendix. MacCullagh's reasoning appears to be so far correct as to have led to correct equations, although through a form of V which may, I conceive, be shown to be inadmissible.

independently of, and in a different manner from Green, precisely Fresnel's laws of double refraction and the accompanying polarization, on the condition, however, that in polarized light the vibrations are parallel to the plane of polarization.

It is remarkable that in the previous year MacCullagh, in a letter to Sir David Brewster*, published expressions for the internal pressures identical with those which result from Green's first theory, provided that in the latter the terms be omitted which arise from that term in which contains μ, a term which vanishes in the case of transversal vibrations propagated within a crystal. It does not appear how these expressions were obtained by MacCullagh; it was probably by a tentative process.

φ

The various theories which have just been reviewed have this one feature in common, that in all, the direct action of the ponderable molecules is neglected, and the ether treated as a single vibrating medium. It was, doubtless, the extreme difficulty of determining the motion of one of two mutually penetrating media that led mathematicians to adopt this, at first sight, unnatural supposition; but the conviction seems by some to have been entertained from the first, and to have forced itself upon the minds of others, that the ponderable molecules must be taken into account in a far more direct manner. Some investigations were made in this direction by Dr. Lloyd as long as twenty-five years agot. Cauchy's later papers show that he was dissatisfied with the method, adopted in his earlier ones, of treating the ether within a ponderable body as a single vibrating medium; but he does not seem to have advanced beyond a few barren generalities, towards a theory of double refraction founded on a calculation of the vibrations of one of two mutually penetrating media. In the theory of double refraction advanced by Professor Challis §, the ether is assimilated to an ordinary elastic fluid, the vibrations of which are modified by resisting masses; and his theory leads him at once to Fresnel's elegant construction of the wave surface by points. The theory, however, rests upon principles which have not received the general assent of mathematicians. In a work entitled "Light explained on the Hypothesis of the Ethereal Medium being a Viscous Fluid", Mr. Moon has put in a clear form some of the more serious objections which may be raised against Fresnel's theory; but that which he has substituted is itself open to formidable objections, some of which the author himself seems to have perceived.

In concluding this part of the subject, I may perhaps be permitted to express my own belief that the true dynamical theory of double refraction has yet to be found.

In the present state of the theory of double refraction, it appears to be of especial importance to attend to a rigorous comparison of its laws with actual observation. I have not now in view the two great laws giving the planes of polarization, and the difference of the squared velocities of propagation, of the two waves which can be propagated independently of each other in any given direction within a crystal. These laws, or at least laws differing from them only by quantities which may be deemed negligible in observation, had previously been discovered by experiment; and the deduction of these laws by Fresnel from his theory, combined with the verification of the law, which his theory, correcting in this respect previous notions, first pointed out, that

*Philosophical Magazine for 1836, vol. viii. p. 103.

+ Proceedings of the Royal Irish Academy, vol. i. p. 10.

See his optical memoirs published in the 22nd volume of the 'Mémoires de l'Académie." § Cambridge Philosophical Transactions, vol. viii. p. 524. Macmillan & Co., Cambridge, 1853.

in each principal plane of a biaxal crystal the ray polarized in that plane obeys the ordinary law of refraction, leaves no reasonable doubt that Fresnel's construction contains the true laws of double refraction, at least in their broad features. But regarding this point as established, I have rather in view a verification of those laws which admit of being put to the test of experiment with extreme precision; for such verifications might often enable the mathematician, in groping after the true theory, to discard at once, as not agreeing with observation, theories which might present themselves to his mind, and on which otherwise he might have spent much fruitless labour.

To make my meaning clearer, I will refer to Fresnel's construction, in which the laws of polarization and wave-velocity are determined by the sections, by a diametral plane parallel to the wave-front, of the ellipsoid*

(11), where a, b, c denote the principal wave-velocities. The principal semiaxes of the section determine by their direction the normals to the two planes of polarization, and by their magnitude the reciprocals of the corresponding wave-velocities. Now a certain other physical theory which might be proposed leads to a construction differing from Fresnel's only in this, that the planes of polarization and wave-velocities are determined by the section, by a diametral plane parallel to the wave-front, of the ellipsoid

the principal semiaxes of the section determining by their direction the normals to the two planes of polarization, and by their magnitudes the corresponding wave-velocities. The law that the planes of polarization of the two waves propagated in a given direction bisect respectively the two supplemental dihedral angles made by planes passing through the wavenormal and the two optic axes, remains the same as before, but the positions of the optic axes themselves, as determined by the principal indices of refraction, are somewhat different; the difference, however, is but small if the differences between a2, b2, c2 are a good deal smaller than the quantities themselves. Each principal section of the wave surface, instead of being a circle and an ellipse, is a circle and an oval, to which an ellipse is a near approximation +. The difference between the inclinations of the optic axes, and between the amounts of extraordinary refraction in the principal planes, on the two theories, though small, are quite sensible in observation, but only on condition that the observations are made with great precision. We see from this example of what great advantage for the advancement of theory observations of this character may be.

One law which admits of receiving, and which has received, this searching comparison with observation, is that according to which, in each principal plane of a biaxal crystal, the ray which is polarized in that plane obeys the ordinary law of refraction, and accordingly in a uniaxal crystal, in which every plane parallel to the axis is a principal plane, the so-called ordinary ray follows rigorously the law of ordinary refraction. This law was carefully verified by Fresnel himself in the case of topaz, by the method of cutting plates parallel to the same principal axis, or axis of elasticity, carefully * It would seem to be just as well to omit the surface of elasticity altogether, and refer the construction directly to the ellipsoid (11).

The equation of the surface of wave-slowness in this and similar cases may be readily obtained by the method given by Professor Haughton in a paper "On the Equilibrium and Motion of Solid and Fluid Bodies," Transactions of the Royal Irish Academy, vol. xxi. p. 172.