B

are joined together along BC, and if a ray traverses such a prism at minimum deviation its direction inside the prism is parallel to the axis. A ray polarized either in the principal plane or at right angles to it, is divided into two rays circularly polarized in oppoc site directions. The same is true therefore of an unpolarized ray. Of these circularly polarized rays one gains over the other while traversing the first prism, and loses equally while passing through the second prism. The combined optical distance is therefore the same for both components, and there is only a single refraction at emergence.

Fig. 172.

A,

164. Crystalline Media. The complete investigation of crystalline substances is complicated and serves no useful purpose, as rotatory effects have only been observed in uniaxal crystals. We may therefore take quartz to be the typical example. Quartz shows no rotatory effects for rays travelling at right angles to the axis. A plane wave travelling in that substance splits up into two plane polarized waves if the wave travels at right angles to the axis, and into two circularly polarized waves if it travels parallel to the axis. In the case of waves travelling obliquely to the axis we may therefore surmise that the two waves are elliptically polarized, the ellipse becoming more and more eccentric as the wave becomes less inclined to the axis. This conclusion is verified by experiment. The elements of the ellipse have been made the subject of calculation by Sir George Airy*. A very clear account of the work of Airy, Jamin and Gouy on this subject is given by Mascart†.

165. Rotatory Dispersion. The rotation per unit length according to (16) is

on the supposition that we need only consider one period 27/n of the free vibration. In this expression ẞ is a constant which can either be positive or negative according to the sign of s. If the free period is very short compared with the range of visible periods, we may neglect w in comparison with n, and the rotation is in that case proportional to w, i.e. inversely proportional to the square of the wave-length. This law holds approximately for most substances which have been examined. In general we have to consider several free periods, so that we must write

the summation having to be carried out for the different values of m. If the free periods lie in the ultra-violet so that all values of nm are larger than we may expand the function in powers of and obtain after rearranging terms

r = r2w2 + r2w1+rzw® .

(18), where r1, 72, 73, are quantities depending on the values of ẞm and nmThe rotatory properties of quartz have been investigated over a very wide range. It is found that the effects may be explained by assuming two ultra-violet free periods, one of which may be made to coincide with the ultra-violet period, which has been deduced from the general dispersion effects of quartz (Art. 150), the other being very short*. The infra-red periods necessary for the explanation of refraction do not seem to produce any rotatory effects.

166. Isochromatic and Achromatic Lines. The appearance of photogyric crystals in the polariscope is materially affected by their rotatory effect. The calculation of the isochromatic and achromatic lines has been carried out by Sir George Airy. A full account is given in Mascart's Optics †. The simplest case is that of a plate cut at right angles to the axis examined with crossed polarizer and analyser. Apart from the rotatory effect, the appearance should be that of Fig. 1, Plate II. Now owing to this rotatory effect the vibration which enters near the centre parallel to the principal plane of the polarizer leaves it inclined at an angle to that direction and is not therefore completely blocked out by the analyser. no achromatic lines near the centre. that of the figure, omitting the dark ring.

The result is that there are The general appearance is cross within the first dark

167. The Zeeman effect. Before discussing the theory of photogyric effects, which a magnetic field impresses on a wave of light passing through it, we may give a short account of the modifications of the luminous radiations observed when the source of light is subjected to strong magnetic forces. It was discovered by Zeeman in 1896 that a sodium flame placed in a magnetic field showed a widening of the two yellow lines, and at the suggestion of H. A. Lorentz, who at once foresaw the right explanation, further experiments were made to test the polarization of the emitted radiations which confirmed Lorentz's theory. In the case of spectroscopic lines, which show the simplest type of magnetic effect, it is found that if the light is examined axially, i.e. parallel to the lines of force, each line splits into two, which are circularly polarized in opposite directions. Looked at

* Drude, Optik, p. 381.

+ Optique, Vol. II. p. 314.

equatorially, each line is divided into three components, the centre one being polarized in an equatorial plane, and the two others in a plane passing through the lines of force.

If we look upon the radiations as being due to the vibrations of an electron these observations admit of a simple explanation. Consider, first, light sent out in the axial direction. Each rectilinear vibration may be supposed to be made up of two opposite circular vibrations, the orbits lying in the equatorial plane. Let the light which reaches the observer travel through the flame in the direction of the lines of force, i.e. from the north to the south magnetic pole. A positive electron performing an anti-clockwise rotation, i.e. a positive rotation round a line of force, will under these circumstances be acted on by a force Hev, tending to increase the diameter of the circle in which it revolves (H-intensity of magnetic field, v = linear speed of electron, e charge of electron). If in the absence of the magnetic field the acceleration is n3d, where d is the displacement, and n'd represents the acceleration in the circular path when the magnetic force acts, we have, p being the

where the upper sign holds for the positive rotation.

=

.(19),

In the case here considered we may write = n, and considering n-n, always to be a small quantity, we may neglect its square. We thus find

Finally, introducing the frequencies N and N, in place of n and n1

The coefficient z may conveniently be called the Zeeman coefficient. We conclude that a rectilinear simply periodic motion is divided into two circular motions, the longer period showing anti-clockwise rotation if e is positive. Zeeman observed that the less refrangible component rotates clockwise, and the more refrangible one anti-clockwise, if the field is in the specified direction, and it follows that if our theory is correct it is the negative electron that gives rise to all vibrations for which this is the case.

Looked at equatorially, the two circular orbits appear in projection as rectilinear vibrations, and send out light vibrating at right angles to the lines of force. So far the vibrations which take place along the lines of force have not been taken into account, but these are not in the simple theory here considered affected by the magnetic field. They constitute therefore plane polarized vibrations transmitted in the equatorial plane having an unmodified period. Looked at equatorially we should expect therefore to see each line divided into three, the external components of the triplet having the same period as the circular vibrations observed in the axial direction. The agreement of the appearance reasoned out in this fashion with the observed facts constitutes a direct proof that the direction of vibration is at right angles to the plane of polarization, identifying variations in electric force with the direction of vibration. Although there is much indirect evidence in favour of this view, such a convincing demonstration as that afforded by the Zeeman effect is very satisfactory.

Our calculation has tacitly assumed that the vibrating electron is free from constraint and acts as an independent unit with three degrees of freedom. If we dropped this assumption we should be led to more complicated magnetic effects, and indeed the majority of spectroscopic lines do not show the simple subdivision which the theory in its simplest form gives us.

H. A. Lorentz* in a general theoretical discussion shows that if a spectroscopic line divides into n components, there must be n degrees of freedom in the system which in the absence of the magnetic field are coincident.

According to the simplest form of the theory, the vibrations parallel to the lines of force preserve their period, but there are important cases in which these also change and two vibrations, one of larger and one of shorter period, take the place of the original one. In some cases the original period is maintained as well; in other cases it completely disappears. Such a phenomenon shows that the vibration along the line of force is not free, but is accompanied by changes in directions at right angles to itself, and that the magnetogyric effect of the accompanying changes reacts on the original vibration.

In view of the importance of the subject, I give a short statement of some of the principal facts which have been established. It is necessary to introduce it by a brief description of the structure of line spectra. Many of the metallic spectra contain a number of lines which form a connected series, and we distinguish between the trunk series (Kayser and Runge's "Hauptserie "), the main-branch series (Kayser

* Rapports présentés au congrès international de Physique de 1900. Vol. II.

page 1.

and Runge's "Zweite Nebenserie"), and the side-branch series (Kayser and Runge's "Erste Nebenserie "). Figure 173 shows diagram

M.B.

S.B.

Fig. 173.

matically the arrangement of lines in the three series, the red end of the spectrum being to the right, the trunk, main branch and side branch being marked T., M.B., S.B., respectively. It is seen that the members of each series approach some definite limit of frequency on the more refrangible side; the point to which each converges I call the "root" of the series. The two branches have a common root at some point in the trunk. According to an important law discovered by Rydberg, and later independently by the author of this book, the frequency of the common root of two branches is obtained by subtracting the frequency of the root of the trunk from that of its least refrangible member. In the spectra of the alkali metals each line of the trunk is a doublet, and we may speak of a twin trunk springing out of the same root. In the same spectra the lines belonging to the two branches are also doublets. The two components of the branch series are not twins springing out of the same root, but the difference in the vibration number is the same for each doublet, there being two roots giving the same difference. Rydberg's law being true for each component of a twin trunk, each set of components of the main branch is associated with one of the two divisions of the trunk, the root of least frequency being attached to that part of the trunk the members of which have the highest frequency. The connexion of the side branches is not so clearly established, there being evidence of a further relation between them and hitherto unknown trunks.

The lines which belong to the branches of the spectra of magnesium, calcium, and of the allied metals occur in triplets, and analogy leads us to think that the trunk must be a triplet also, each of the three compounds having one main and one side branch. The trunk vibrations however have not been seen or identified in those metals.

Passing on to the behaviour of the different vibrations in the magnetic field, it was first announced by Preston, but more particularly proved