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quantity (u’— 1)/(u2+1), in which y represents the refractive index of the pressural waves or a quantity of that nature which is to be made nearly equal to unity, cannot on any rigid theory be accepted. Hence neither Green's theory, nor any other rigid elastic solid theory, such as that of Voigt or of K. Pearson, can be made to agree with experiment, if we suppose the ether incompressible. The alternative assumption of supposing the ether contractile, will be found to agree with experiment, but there is no means of deciding between it and the electromagnetic theory as regards refraction and reflection,

These theories lead to the following expressions for the ratio of the intensities and the difference of phase of the components, polarised parallel and perpendicular to the plane of incidence, of the reflected light

cos? (1,+i) sin i, sin is cos i, cos i, cos (io+) RII cos? (-;)

cos' (1-2)

sin’i, sino i, cos i, cos i,

cos (10-2) tan (pl-pll)

E sin’i, cos io
cos (10 — ) cos(io +ii) (1+ Do cosio) +DE sinoi, cosin

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where Rd, R// are the amplitudes, pl, pll the phase retardations of the components polarised perpendicularly (1) and parallel (II) to the plane of incidence, the amplitude of each incident component being taken as unity, io, i, are the angles of incidence and refraction, and A, B, D, E are four constants depending on the refractive indices of the two media and on the constitution of the variable layer and satisfying the theoretical conditions B = (M./Mo) E®, and the conditions that when the two media are inverted, that is, Mo, Mi interchanged, A, B remain unchanged, and M. E change sign.

The expressions for (RI/R//)and for tan (p1-pll) show that the change of difference of phase depends chiefly on E, being also slightly modified by D, and that the alteration in amplitude depends chiefly on B, being only slightly affected by A. The accuracy with which the constants can be experimentally determined is thus very different, being greatest for E, less for B, and least for D and A.

The values of the four constants are independently calculated for several pairs of media from experiments by Jamin, by Kurz, and by Quincke, using the method of least squares and assuming that the observations are equally exact at all incidences, which is only roughly true.

The numbers for the ratio intensities and the difference of phase calculated with the values of the constants agree rather better with

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observation than the corresponding numbers for Cauchy's formulæ, even as modified by Quincke to explain his experiments. For inEtance, in the case of Jamin’s experiments on realgar-air, chosen at random, the probable errors of a single observation are-according to theory, in tan-(R1/R/I), 20-82, according to Cauchy, 22:96, and in the difference of phase pt-pll, measured in wave-lengths, according to the theory, 0·0054, according to Cauchy, 0·0071. This superiority is due to the influence of the additional constants D and A; but their effect is very slight, that of A especially so; in some cases A may be put zero without greatly impairing the accuracy of the theoretical formulæ.

The values of B, as determined from the intensity experiments agree within the limits of error with those deduced from the equation

(41/41) E?; in the one case of essence of lavender-air there is a serious discrepancy, for B is 0.000027, while (mi/10) E’ is 0·000065; but here the error is due to the smallness of B, ten times less than for any other pair of media; in fact the minimum value of (R1/R/I)2 is only tan7' or 0:000004, and an error of only a few minntes in the determination of the azimuth would make a great difference in the value of B. In cases of such difference between B and (ui/mo)E?, it is best to rely on the value of E, which can be determined with much greater accuracy than B, in some cases with more than five times greater accuracy.

As regards the interchange of media, there were available three sufficiently accurate sets of experiments by Quincke. The conditions referred to above are satisfied with very good accuracy in the case of flintglass-air, the values of B being 0.00533 and 0.0050, and of u E,

- (UE)', 0.0925, 0.093; for flintglass-water the numbers are, B, 0.0120, 0·0100, and 4 E, 0:1490, –(UE)', 0:1426; but for crownglass-air the discrepancy is great, B, 0.00040, 0.00111, and ME, 0.0237, -(4E)', 0:0427.

Jamin and Cauchy's corresponding relation between the ellipticities, –E'/E = pilni', is not so nearly satisfied; the observed values of - E'/E are in the three cases, 1.741, 1.244, and 3:446, instead of 1.616, 1:210, and 1.515. More accurate experiments will be necessary to test this point. Cauchy's theory leads to a relation between the ellipticities of

E2 E23 E31 three media taken two and two, + + = 0, which relation has

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Ma been tested by Quincke and found not at all in accordance with experiment.

The above theory has the advantage in requiring no such relation.

The theory discussed above merely requires the existence of a transition film, whether due to actual transition between the ether of the two media or to an adventitious accumulation of dust,

moisture, or condensed gases, or to combinations of these causes. And it affords an explanation of the details of reflection, which is rigid, and at least as good as the representation given by the empirical formulæ of Cauchy, even as modified by Quincke.

VI. “On the Transformation of Optical Wave-Surfaces by

Homogeneous Strain.” By OLIVER HEAVISIDE, F.R.S.
Received December 20, 1893.

Simplex Eolotropy. 1. All explanations of double refraction (proximate, not ultimate) rest upon the hypothesis that the medium in which it occurs is so structured as to impart eolotropy to one of the two properties, associated with potential and kinetic energy, with which the ether is endowed in order to account for the transmission of waves through it in the simplest manner. It may be elastic eolotropy, or it may be something equivalent to eolotropy as regards the density. In Maxwell's electromagnetic theory the two properties are those connecting the electric force with the displacement, and the magnetic force with the induction, say the permittivity and the inductivity, or c and u. These are, in the simplest case, constants corresponding to isotropy. The existence of eolotropy as regards either of them will cause double refraction. Then either c or u is a symmetrical linear operator, or dyadic, as Willard Gibbs calls it. In either case the optical wavesurface is of the Fresnel type. In either case the fluxes displacement and induction are perpendicular to one another and in a wave-front, whilst the electric and magnetic forces are also perpendicular to one another. But it is the magnetic force that is in the wave-front, coincident with the induction, in

case of magnetic isotropy and electric eolotropy, the electric force being then out of the wave-front, though in the plane of the normal and the displacement. And in the other extreme case of electric isotropy and magnetic eolotropy, the electric force is in the wavefront, coincident with the displacement, whilst the magnetic force is out of the wave-front, thongh in the plane of the normal and the induction. Now, as a matter of fact, crystals may be strongly eolotropic electrically, whilst their magnetic eolotropy, if existent, is insignificant. This, of course, justifies Maxwell's ascription of double refraction to electric eolotropy.

Properties connected with Duplex Eolotropy. 2. When duplex eolotropy, electric and magnetic, is admitted, we obtain a more general kind of wave-surface, including the former two as extreme cases. It is almost a pity that magnetic eolotropy should be insensible, because the investigation of the conditions regulating plane waves in media possessing duplex eolotropy, and the wavesurface associated therewith, possesses many points of interest. The chief attraction lies in the perfectly symmetrical manner in which the subject may be displayed, as regards the two eolotropies. This brings out clearly properties which are not always easily visible in the case of simplex eolotropy, when there is a one-sided and imperfect development of the analysis concerned.

In general, the fluxes displacement and induction, although in the wave-front, are not coperpendicular. Corresponding to this, the two forces electric and magnetic, which are always in the plane perpendicular to the ray, or the fax of energy, are not coperpendicular. Nor are the positions of the fluxes in the wave-front conditioned by the effective components in that plane of the forces being made to coincide with the Auxes. There are two waves with a given normal, and it would be impossible to satisfy this requirement for both. But there is a sort of balance of skewness, inasmuch as the positions of the fluxes in the wave-front are such that the angle through which the plane containing the normal and the displacement (in either wave) must be turned, round the normal as axis, to reach the electric force, is equal (though in the opposite sense) to the angle through which the plane containing the normal and the induction must be turned to reach the magnetic force. These are merely rudimentary properties. I have investigated the wave-surface and associated matters in my paper “On the Electromagnetic Wave-surface(* Pbil. Mag.,' June, 1885; or Electrical Papers,' vol. 2, p. 1).

Effects of straining a Duplex Wave-surface. 3. The connexion between the simplex and duplex types of wavesurface has been interestingly illustrated lately by Dr. J. Larmor in his paper “On the Singularities of the Optical Wave-surface,” (Proc. London Math. Soc.,' vol. 24, 1893). He points out, incidentally, that a simplex wave-surface, when subjected to a particular sort of homogeneous strain, becomes a duplex wave-surface of a special kind. To more precisely state the connexion, let there be electric eolotropy, say c, with magnetic isotropy. Then, if the strainer, or strain operator, applied to the simplex wave-surface, be homologous with c, given by c* x constant, the result is to turn it into a duplex wave-surface whose two eolotropies are also homologous with the original c; that is to say, the principal axes are parallel. This duplex wave-surface is, of course, of a specially simplified kind, though not the simplest. That occurs when the two eolotropies are not merely homologous, but are in constant ratio. The wave-surface then reduces to a single ellipsoid.

Conversely, therefore, if we start with the duplex wave-surface corresponding to homologous permittivity and inductivity, and homogeneously strain it, the strainer being proportional to c', we convert it to a simplex wave-surface whose one eolotropy is homologous with the former two.

Remembering that the equation of the duplex wave-surface is symmetrical with respect to the two colotropies, so that they may be interchanged without altering the surface, it struck me on reading Dr. Larmor's remarks that a similar reduction to a simplex wavesurface could be effected by a strainer proportional to mest. This was verified on examination, and some more general transformations presented themselves. The results are briefly these :

Any duplex wave-surface (irrespective of homology of eolotropies), when subjected to homogeneous strain (not necessarily pure), usually remains a duplex wave-surface. That is, the transformed surface is of the same type, though with different inductivity and permittivity operators.

But in special cases it becomes a simplex wave-surface. In one way the strainer is c/[c], where the square brackets indicate the determinant of the enclosed operator. In another the strainer is M*/[r]. These indicate the strain operator to be applied to the vector of the old surface to produce that of the new one.

Now, these simplex wave-surfaces may be strained anew to their reciprocals with respect to the unit sphere, or the corresponding index-surfaces, which are surfaces of the same type. So we have at least four ways of straining any duplex wave-surface to a simplex one.

Furthermore, any duplex wave-surface may be homogeneously strained to its reciprocal, the corresponding index-surface, of the same duplex type. The strain is pure, but is complicated, as it involves both o and

M.

The strainer is c-'(cu-), divided by the determinant of the same. This transformation is practically the generalization for the duplex wave-surface of Plücker's theorem relating to the Fresnel surface, for that also involves straining the wavesurface to its reciprocal.

Instead of the single strain above mentioned, we may employ three successive

pure strains. Thus, first strain the duplex wave-surface to a simplex surface. Secondly, strain the latter to its reciprocal. Thirdly, strain the last to the reciprocal of the original duplex wavesurface. There are at least two sets of three successive strains which effect the desired transformation. The investigation follows.

Forms of the Index- and Wave-surface Equations, and the Properties of

Inversion and Interchangeability of Operators. 4. Let the electric and magnetic forces be E and H, and the

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