two "indices corre

But in quartz and a

with the spherical wave-shell; or the spond in one direction of the ray. few other crystals, as we shall hereafter find, there is a peculiar kind of double refraction in the direction of the optic axis itself, so that the spheroid does not reach to the spherical wave-shell, but is altogether contained within it. In this case also, however, the optic axis is a common perpendicular to parallel planes which are opposite tangentplanes to the spherical and spheroidal wave-shells.

125. Bi-axial Crystals. Sir David Brewster discovered that there were many other crystals in which neither ray obeyed the law of ordinary refraction; and further experiment showed that such crystals possessed two axes of no double refraction. The phenomena of these crystals are explained on similar principles, but must be studied more in detail later on. In them also the optic axes are the perpendiculars to planes, which are common tangent-planes to the two wave-shells.

126. Phenomena of the Tourmalines.-We can now readily understand the appearances presented by the tourmalines and other apparatus. Considering the original beam of common light to consist either of vibrations in all azimuths, as at A, Fig. 137, or only of two at right angles with each other, as at B in the same figure,1 which may, however, be at any angle with the horizon; in either case polarisation consists in the obtaining separately of vibrations in one definite path only; and "plane" polarisation (for we

1 Either hypothesis will account for most of the phenomena, and with the two double-image prisms in one position, we did certainly compound one beam (F, Fig. 132) out of two rectangular plane-polarised beams, which cannot be distinguished by any test yet known from common light. Nevertheless the theory that common light contains all azimuths is far the most probable, and best meets certain important considerations. A brief summary of the matter is given in the Appendix to this chapter.

shall find other kinds) means that such path is in a plane. The phenomena then are very easily explained. Taking the tourmalines as an example, the original ray a (considered for convenience as compounded of two planes, or polarised rays, together) meets the first tourmaline, B (Fig. 146), so placed that the optic axis is perpendicular. We have already learnt that the ray must be divided into planes of vibration, one perpendicular and the other horizontal: but the horizontal vibrations are all absorbed within the crystal, which from some arrangement of its molecules not understood, is (when of a certain thickness) absolutely opaque

B

C

FIG. 146.-Action of Tourmalines.

to them. Therefore only perpendicular vibrations can get through. The ray which emerges, C, is therefore polarised, and if it meets a second tourmaline similarly placed, it will again get through. But supposing the second be placed as at D, all light must be stopped, as we have seen it is, though by two nearly transparent crystals!

The phenomena of the double-image prisms are explained in precisely the same way, except that in this case both rays get through the first crystal. When the second prism is at right angles with the first, the ray that got through the first would be quenched by the same plane of

vibration in the second; but the plane at. right angles with that allows it free passage. The phenomena of reflection and refraction do not need any further analysis; and it will only be necessary, before proceeding with our experiments, to describe briefly the most convenient polarising apparatus.

APPENDIX TO CHAPTER X.

The Vibrations of Common Light.

We have only been able to account for the phenomena of polarised light on the supposition that the particles of ether moved in perfectly definite orbits; and every experiment in this branch of physical optics, from first to last, confirms that hypothesis, which we shall see has also enabled most marvellous predictions to be made, afterwards verified by experiment. But when we proceed to ask, What are the nature or orbits of the vibrations in a ray of common or unpolarised light, we have propounded a question the answer to which is by no means easy. All that can be done here is to state briefly the chief results of the investigations various physicists have made into this interesting subject, and finally to suggest what appears the most probable method of reconciling the chief difficulties.

1

If common light passes through a doubly-refracting film it is divided into two plane-polarised beams; and if these are not much separated, or supposing they are, if

1 It is right to state that great part of the following paragraphs are mainly a condensation, recast, of the admirable summary of the subject given in the last edition of Müller-Pouillet's Lehrbuch der Physik. A translation, or version in any sense, is not pretended; and the conclusion is not from that work.

they are again united, the two together behave as common light (§ 126). This may be and has been accounted for on the theory that common light itself consists of two planepolarised beams, the vibrations of which are in rectilinear planes, so that a section of these planes across the ray would resemble a in Fig. 147. This seems to have been the belief of Sir David Brewster, and partly of Fresnel.

It

But this theory presents great mechanical difficulties. is difficult to conceive that in any homogeneous transparent medium there should be any two given planes of vibration more than any others. Still more, either half of the ether particles vibrate in one plane, and half in the other, or else the same particle alternately vibrates in opposite planes.

Х

B

D

E

FIG. 147. Theories concerning Common Light.

The last supposition cannot possibly be entertained; but even upon the other, immediately contiguous particles of ether must be vibrating in opposite planes. This is difficult to explain on any wave-theory, which supposes that each moving particle communicates similar motions to contiguous particles. And lastly, such a theory seems to suppose that there is really no unpolarised light at all, but solely bundles made up of oppositely polarised rays.

Fresnel therefore adopted the theory that the vibrations of the ether particles in common light took place in all azimuths, but that these azimuths were assumed in succession. On this hypothesis, taking for clearness only a few azimuths, a section of the ray may be supposed to be like B, Fig. 147.

In adopting or describing this theory it is often added,' that with such inconceivably rapid vibrations as those of light it is possible or probable that hundreds or even thousands of vibrations may take place in one plane, before they change into some other plane. But it is impossible, on reflection, to rest satisfied with any such supposition. If the vibrations thus remain constant in direction through any number, there is no reason why their direction should then change; and in fact whenever we have, as in a polarised ray of any kind, one which we know vibrates many times in succession in one orbit, we also know that such orbit of vibration remains stable.

Other considerations also require us to carry our reasoning further. If there be change, it must be either sudden (or by steps, as it were) or by insensible degrees. Obvious mechanical reasons are against the first supposition, and we are almost shut up, therefore, to the hypothesis of a gradual and continuous change in the path of vibration. But even if we suppose the elementary forms of such paths to be plane vibrations, such a gradual change must result in a curve; and when we remember that an elliptical orbit is of all forms of polarisation the most common in nature,2 we are almost driven to take elliptical orbits into our purview. In fact Fresnel's own conception has been modified so as to consist of elliptical vibrations in various azimuths, resembling those of c, Fig. 147, rather than the plane orbits of B in the same figure. Such vibrations gradually changing azimuth would somewhat resemble the curve shown in D, Fig. 147. But the case is not only conceivable, but on every mechanical ground far the most probable, that any

1 See Polarisation of Light, by W. Spottiswoode, P.R.S. (Nature Series), p. 6. See also Deschanel's Natural Philosophy, last page. 2 Elliptical and circular polarisation are explained in a subsequent chapter.