# 1911 Encyclopædia Britannica/Magneto-Optics

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​MAGNETO-OPTICS. The first relation between magnetism and light was discovered by Faraday,$^1$ who proved that the plane of polarization of a ray of light was rotated when the ray travelled through certain substances parallel to the lines of magnetic force. This power of rotating the plane of polarization in a magnetic field has been shown to be possessed by all refracting substances, whether they are in the solid, liquid or gaseous state. The rotation by gases was established independently by H. Becquerel,$^2$ and Kundt and Röntgen,$^3$ while Kundt$^4$ found that films of the magnetic metals, iron, cobalt, nickel, thin enough to be transparent, produced enormous rotations, these being in iron and cobalt magnetized to saturation at the rate of 200,000° per cm. of thickness, and in nickel about 89,000°. The direction of rotation is not the same in all bodies. If we call the rotation positive when it is related to the direction of the magnetic force, like rotation and translation in a right-handed screw, or, what is equivalent, when it is in the direction of the electric currents which would produce a magnetic field in the same direction as that which produces the rotation, then most substances produce positive rotation. Among those that produce negative rotation are ferrous and ferric salts, ferricyanide of potassium, the salts of lanthanum, cerium and didymium, and chloride of titanium.$^5$
The magnetic metals iron, nickel, cobalt, the salts of nickel and cobalt, and oxygen (the most magnetic gas) produce positive rotation.
For slightly magnetizable substances the amount of rotation in a space PQ is proportional to the difference between the magnetic potential at P and Q; or if θ is the rotation in PQ, Ω$_P$, Ω$_Q$, the magnetic potential at P and Q, then θ = R(Ω$_P$ − Ω$_Q$), where R is a constant, called Verdet’s constant, which depends upon the refracting substance, the wave length of the light, and the temperature. The following are the values of R (when the rotation is expressed in circular measure) for the D line and a temperature of 18° C.:—
| Substance. | R × 10$^5$. | Observer. |
| Carbon bisulphide | {1.222 | Lord Rayleigh$^6$ and Köpsel.$^7$ |
| Carbon bisulphide | {1.225 | Rodger and Watson.$^8$ |
| Water | {.377 | Arons.$^9$ |
| Water | {.3808 | Rodger and Watson.$^8$ |
| Alcohol | .330 | Du Bois.$^{10}$ |
| Ether | .315 | Du Bois.$^{10}$ |
| Oxygen (at 1 atmosphere) | .000179 | Kundt and Röntgen (loc. cit.) |
| Faraday’s heavy glass | 1.738 | |
​ The variation of Verdet’s constant with temperature has been determined for carbon bisulphide and water by Rodger and Watson (loc. cit.). They find if R$_t$, R$_0$ are the values of Verdet’s constant at t°C and 0°C. respectively, then for carbon bisulphide R$_t$ = R$_0$ (1 − .0016961), and for water R$_t$ = R$_0$ (1 − .0000305t − .00000305t $^2$).
For the magnetic metals Kundt found that the rotation did not increase so rapidly as the magnetic force, but that as this force was increased the rotation reached a maximum value. This suggests that the rotation is proportional to the intensity of magnetization, and not to the magnetic force.
The amount of rotation in a given field depends greatly upon the wave length of the light; the shorter the wave length the greater the rotation, the rotation varying a little more rapidly than the inverse square of the wave length. Verdet$^{11}$ has compared in the cases of carbon bisulphide and creosote the rotation given by the formula
| θ = mcγ | c$^2$ | ( c − λ | di | ) |
| θ = mcγ | λ$^2$ | ( c − λ | dλ | ) |
with those actually observed; in this formula θ is the angular rotation of the plane of polarization, m a constant depending on the medium, λ the wave length of the light in air, and i its index of refraction in the medium. Verdet found that, though the agreement is fair, the differences are greater than can be explained by errors of experiment.
Verdet$^{12}$ has shown that the rotation of a salt solution is the sum of the rotations due to the salt and the solvent; thus, by mixing a salt which produces negative rotation with water which produces positive rotation, it is possible to get a solution which does not exhibit any rotation. Such solutions are not in general magnetically neutral. By mixing diamagnetic and paramagnetic substances we can get magnetically neutral solutions, which, however, produce a finite rotation of the plane of polarization. The relation of the magnetic rotation to chemical constitution has been studied in great detail by Perkin,$^{13}$ Wachsmuth,$^{14}$ Jahn$^{15}$ and Schönrock.$^{16}$
The rotation of the plane of polarization may conveniently be regarded as denoting that the velocity of propagation of circular-polarized light travelling along the lines of magnetic force depends upon the direction of rotation of the ray, the velocity when the rotation is related to the direction of the magnetic force, like rotation and translation on a right-handed screw being different from that for a left-handed rotation. A plane-polarized ray may be regarded as compounded of two oppositely circularly-polarized rays, and as these travel along the lines of magnetic force with different velocities, the one will gain or lose in phase on the other, so that when they are again compounded they will correspond to a plane-polarized ray, but in consequence of the change of phase the plane of polarization will not coincide with its original position.
Reflection from a Magnet.—Kerr$^{17}$ in 1877 found that when plane-polarized light is incident on the pole of an electromagnet, polished so as to act like a mirror, the plane of polarization of the reflected light is rotated by the magnet. Further experiments on this phenomenon have been made by Righi,$^{18}$ Kundt,$^{19}$ Du Bois,$^{20}$ Sissingh,$^{21}$ Hall,$^{22}$ Hurion,$^{23}$ Kaz$^{24}$ and Zeeman.$^{25}$ The simplest case is when the incident plane-polarized light falls normally on the pole of an electromagnet. When the magnet is not excited the reflected ray is plane-polarized; when the magnet is excited the plane of polarization is rotated through a small angle, the direction of rotation being opposite to that of the currents exciting the pole. Righi found that the reflected light was slightly elliptically polarized, the axes of the ellipse being of very unequal magnitude. A piece of gold-leaf placed over the pole entirely stops the rotation, showing that it is not produced in the air near the pole. Rotation takes place from magnetized nickel and cobalt as well as from iron, and is in the same direction (Hall). Righi has shown that the rotation at reflection is greater for long waves than for short, whereas, as we have seen, the Faraday rotation is greater for short waves than for long. The rotation for different coloured light from iron, nickel, cobalt and magnetite has been measured by Du Bois; in magnetite the direction of rotation is opposite to that of the other metals. When the light is incident obliquely and not normally on the polished pole of an electromagnet, it is elliptically polarized after reflection, even when the plane of polarization is parallel or at right angles to the plane of incidence. According to Righi, the amount of rotation when the plane of polarization of the incident light is perpendicular to the plane of incidence reaches a maximum when the angle of incidence is between 44° and 68°, while when the light is polarized in the plane of incidence the rotation steadily decreases as the angle of incidence is increased. The rotation when the light is polarized in the plane of incidence is always less than when it is polarized at right angles to that plane, except when the incidence is normal, when the two rotations are of course equal.
Reflection from Tangentially Magnetized Iron.—In this case Kerr$^{26}$ found: (1) When the plane of incidence is perpendicular to the lines of magnetic force, no rotation of the reflected light is produced by magnetization; (2) no rotation is produced when the light is incident normally; (3) when the incidence is oblique, the lines of magnetic force being in the plane of incidence, the reflected light is elliptically polarized after reflection, and the axes of the ellipse are not in and at right angles to the plane of incidence. When the light is polarized in the plane of incidence, the rotation is at all angles of incidence in the opposite direction to that of the currents which would produce a magnetic field of the same sign as the magnet. When the light is polarized at right angles to the plane of incidence, the rotation is in the same direction as these currents when the angle of incidence is between 0° and 75° according to Kerr, between 0° and 80° according to Kundt, and between 0° and 78° 54′ according to Righi. When the incidence is more oblique than this, the rotation of the plane of polarization is in the opposite direction to the electric currents which would produce a magnetic field of the same sign.
The theory of the phenomena just described has been dealt with by Airy,$^{27}$ C. Neumann,$^{28}$ Maxwell,$^{29}$ Fitzgerald,$^{30}$ Rowland,$^{31}$ H. A. Lorentz,$^{32}$ Voight,$^{33}$ Ketteler,$^{34}$ van Loghem,$^{35}$ Potier,$^{36}$ Basset,$^{37}$ Goldhammer,$^{38}$ Drude,$^{39}$ J. J. Thomson,$^{40}$ and Leatham;$^{41}$ for a critical discussion of many of these theories we refer the reader to Larmor’s$^{42}$ British Association Report. Most of these theories have proceeded on the plan of adding to the expression for the electromotive force terms indicating a force similar in character to that discovered by Hall (see Magnetism) in metallic conductors carrying a current in a magnetic field, i.e. an electromotive force at right angles to the plane containing the magnetic force and the electric current, and proportional to the sine of the angle between these vectors. The introduction of a term of this kind gives rotation of the plane of polarization by transmission through all refracting substance, and by reflection from magnetized metals, and shows a fair agreement between the theoretical and experimental results. The simplest way of treating the questions seems, however, to be to go to the equations which represent the propagation of a wave travelling through a medium containing ions. A moving ion in a magnetic field will be acted upon by a mechanical force which is at right angles to its direction of motion, and also to the magnetic force, and is equal per unit charge to the product of these two vectors and the sine of the angle between them. For the sake of brevity we will take the special case of a wave travelling parallel to the magnetic force in the direction of the axis of z.
Then supposing that all the ions are of the same kind, and that there are n of these each with mass m and charge e per unit volume, the equations representing the field are (see Electric Waves):—
| K$_0$ | dX$_0$ | + 4πne | dξ | = | dβ | ; |
| K$_0$ | dt | + 4πne | dt | = | dz | ; |
| dX$_0$ | = | dβ | ; |
| dz | = | dt | ; |
| K$_0$ | dY$_0$ | + 4πne | dη | = − | dα |
| K$_0$ | dt | + 4πne | dt | = − | dz |
| dY$_0$ | = − | dα | ; |
| dz | = − | dt | ; |
| m | d $^2$ξ | + R$_1$ | dξ | + aξ = ( X$_0$ + | 4π | neξ ) e + He | dη |
| m | dt $^2$ | + R$_1$ | dt | + aξ = ( X$_0$ + | 3 | neξ ) e + He | dt |
| m | d $^2$η | + R$_1$ | dη | + aη = ( Y$_0$ + | 4π | neη ) e − He | dξ | ; |
| m | dt $^2$ | + R$_1$ | dt | + aη = ( Y$_0$ + | 3 | neη ) e − He | dt | ; |
where H is the external magnetic field, X$_0$, Y$_0$ the components of the part of the electric force in the wave not due to the charges on the atoms, α and β the components of the magnetic force, ξ and η ​the co-ordinates of an ion, R$_1$ the coefficient of resistance to the motion of the ions, and α the force at unit distance tending to bring the ion back to its position of equilibrium, K$_0$ the specific inductive capacity of a vacuum. If the variables are proportional to ε$^{l(pt−qz)}$ we find by substitution that q is given by the equation
| q$^2$ − K$_0$p$^2$ − | 4πne$^2$p$^2$P | = ± | 4πne$^3$Hp$^3$ | , |
| q$^2$ − K$_0$p$^2$ − | P$^2$ − H$^2$e$^2$p$^2$ | = ± | P$^2$ − H$^2$e$^2$p$^2$ | , |
or, by neglecting R, P = m (s$^2$ − p$^2$), where s is the period of the free ions. If, q$_1$$^2$, q$_2$$^2$ are the roots of this equation, then corresponding to q$_1$ we have X$_0$ = ιY$_0$ and to q$_2$ X$_0$ = −ιY$_0$. We thus get two oppositely circular-polarized rays travelling with the velocities p/q$_1$ and p/q$_2$ respectively. Hence if v$_1$, v$_2$ are these velocities, and v the velocity when there is no magnetic field, we obtain, if we neglect terms in H$^2$,
| 1 | = | 1 | + | 4πne$^3$Hp | , |
| v$_1$$^2$ | = | v$^2$ | + | m$^2$ (s$^2$ − p$^2$)$^2$ | , |
| 1 | = | 1 | − | 4πne$^3$Hp | . |
| v$_2$$^2$ | = | v$^2$ | − | m$^2$ (s$^2$ − p$^2$)$^2$ | . |
The rotation r of the plane of polarization per unit length
| = 1/2p ( | 1 | − | 1 | ) = | 2πne$^3$Hp$^2$v | . |
| = 1/2p ( | v$_1$ | − | v$_2$ | ) = | m$^2$ (s$^2$ − p$^2$)$^2$ | . |
Since 1/v$^2$ = K$_0$ + 4πne$^2$/m (s$^2$ − p$^2$), we have if µ is the refractive index for light of frequency p, and v$_0$ the velocity of light in vacuo.
| µ$^2$ − 1 = 4πne$^2$v$_0$$^2$/m (s$^2$ − p$^2$). | (1) |
So that we may put
| r = (µ$^2$ − 1)$^2$ p$^2$H / sπµnev$_0$$^3$. | (2) |
Becquerel (Comptes rendus, 125, p. 683) gives for r the expression
| 1/2 | e | H | dµ | , |
| 1/2 | m | v$_0$ | dλ | , |
where λ is the wave length. This is equivalent to (2) if µ is given by (1). He has shown that this expression is in good agreement with experiment. The sign of r depends on the sign of e, hence the rotation due to negative ions would be opposite to that for positive. For the great majority of substances the direction of rotation is that corresponding to the negation ion. We see from the equations that the rotation is very large for such a value of p as makes P = 0; this value corresponds to a free period of the ions, so that the rotation ought to be very large in the neighbourhood of an absorption band. This has been verified for sodium vapour by Macaluso and Corbino.$^{43}$
If plane-polarized light falls normally on a plane face of the medium containing the ions, then if the electric force in the incident wave is parallel to x and is equal to the real part of Aεl(pt−qz), if the reflected beam in which the electric force is parallel to x is represented by Bεl(pt+qz) and the reflected beam in which the electric force is parallel to the axis of y by Cεl(pt+qz), then the conditions that the magnetic force parallel to the surface is continuous, and that the electric forces parallel to the surface in the air are continuous with Y$_0$, X$_0$ in the medium, give
| A | = | B | = | ιC |
| (q + q$_1$) (q + q$_2$) | = | (q$^2$ − q$_1$q$_2$) | = | q (q$_2$ − q$_1$) |
or approximately, since q$_1$ and q$_2$ are nearly equal,
| ιC | = | q (q$_2$ − q$_1$) | = | (µ$^2$ − 1) pH | . |
| B | = | q$^2$ − q$_1$$^2$ | = | 4πµneV$_0$$^2$ | . |
Thus in transparent bodies for which µ is real, C and B differ in phase by π/2, and the reflected light is elliptically polarized, the major axis of the ellipse being in the plane of polarization of the incident light, so that in this case there is no rotation, but only elliptic polarization; when there is strong absorption so that µ contains an imaginary term, C/B will contain a real part so that the reflected light will be elliptically polarized, but the major axis is no longer in the plane of polarization of the incident light; we should thus have a rotation of the plane of polarization superposed on the elliptic polarization.
Zeeman’s Effect.—Faraday, after discovering the effect of a magnetic field on the plane of polarization of light, made numerous experiments to see if such a field influenced the nature of the light emitted by a luminous body, but without success. In 1885 Fievez,$^{44}$ a Belgian physicist, noticed that the spectrum of a sodium flame was changed slightly in appearance by a magnetic field; but his observation does not seem to have attracted much attention, and was probably ascribed to secondary effects. In 1896 Zeeman$^{45}$ saw a distinct broadening of the lines of lithium and sodium when the flames containing salts of these metals were between the poles of a powerful electromagnet; following up this observation, he obtained some exceedingly remarkable and interesting results, of which those observed with the blue-green cadmium line may be taken as typical. He found that in a strong magnetic field, when the lines of force are parallel to the direction of propagation of the light, the line is split up into a doublet, the constituents of which are on opposite sides of the undisturbed position of the line, and that the light in the constituents of this doublet is circularly polarized, the rotation in the two lines being in opposite directions. When the magnetic force is at right angles to the direction of propagation of the light, the line is resolved into a triplet, of which the middle line occupies the same position as the undisturbed line; all the constituents of this triplet are plane-polarized, the plane of polarization of the middle line being at right angles to the magnetic force, while the outside lines are polarized on a plane parallel to the lines of magnetic force. A great deal of light is thrown on this phenomenon by the following considerations due to H. A. Lorentz.$^{46}$
Let us consider an ion attracted to a centre of force by a force proportional to the distance, and acted on by a magnetic force parallel to the axis of z: then if m is the mass of the particle and e its charge, the equations of motion are
| m | d $^2$x | + αx = − He | dy | ; |
| m | dt$^2$ | + αx = − He | dt | ; |
| m | d $^2$y | + αy = He | dx | ; |
| m | dt$^2$ | + αy = He | dt | ; |
| m | d $^2$z | + ax = 0. |
| m | dt$^2$ | + ax = 0. |
The solution of these equations is
* x = A cos (p$_1$t + β) + B cos (p$_2$t + β$_1$)
* y = A sin (p$_1$t + β) − B sin (p$_2$t + β$_1$)
* z = C cos (pt + γ)
or approximately
| p$_1$ = p + 1/2 | He | ,   p$_2$ = p − 1/2 | He | . |
| p$_1$ = p + 1/2 | m | ,   p$_2$ = p − 1/2 | m | . |
Thus the motion of the ion on the xy plane may be regarded as made up of two circular motions in opposite directions described with frequencies p$_1$ and p$_2$ respectively, while the motion along z has the period p, which is the frequency for all the vibrations when H = 0. Now suppose that the cadmium line is due to the motion of such an ion; then if the magnetic force is along the direction of propagation, the vibration in this direction has its period unaltered, but since the direction of vibration is perpendicular to the wave front, it does not give rise to light. Thus we are left with the two circular motions in the wave front with frequencies p$_1$ and p$_2$ giving the circularly polarized constituents of the doublet. Now suppose the magnetic force is at right angles to the direction of propagation of the light; then the vibration parallel to the magnetic force being in the wave front produces luminous effects and gives rise to a plane-polarized ray of undisturbed period (the middle line of the triplet), the plane of polarization being at right angles to the magnetic force. The components in the wave-front of the circular orbits at right angles to the magnetic force will be rectilinear motions of frequency p$_1$ and p$_2$ at right angles to the magnetic force—so that they will produce plane-polarized light, the plane of polarization being parallel to the magnetic force; these are the outer lines of the triplet.
If Zeeman’s observations are interpreted from this point of view, the directions of rotation of the circularly-polarized light in the doublet observed along the lines of magnetic force show that the ions which produce the luminous vibrations are negatively electrified, while the measurement of the charge of frequency due to the magnetic field shows that e/m is of the order 10$^7$. This result is of great interest, as this is the order of the value of e/m in the negatively electrified particles which constitute the Cathode Rays (see Conduction, Electric III. Through Gases). Thus we infer that the “cathode particles” are found in bodies, even where not subject to the action of intense electrical fields, and are in fact an ordinary constituent of the molecule. Similar particles are found near an incandescent wire, and also near a metal plate illuminated by ultra-violet light. The value of e/m deduced from the Zeeman effect ranges from 10$^7$ to 3.4 × 10$^7$, the value of e/m for the particle in the cathode rays is 1.7 × 10$^7$. The majority of the determinations of e/m from the Zeeman effect give numbers larger than this, the maximum being about twice this value.
​A more extended study of the behaviour of the spectroscopic lines has afforded examples in which the effects produced by a magnet are more complicated than those we have described, indeed the simple cases are much less numerous than the more complex. Thus Preston$^{47}$ and Cornu$^{48}$ have shown that under the action of a transverse magnetic field one of the D lines splits up into four, and the other into six lines; Preston has given many other examples of these quartets and sextets, and has shown that the change in the frequency, which, according to the simple theory indicated, should be the same for all lines, actually varies considerably from one line to another, many lines showing no appreciable displacement. The splitting up of a single line into a quartet or sextet indicates, from the point of view of the ion theory, that the line must have its origin in a system consisting of more than one ion. A single ion having only three degrees of freedom can only have three periods. When there is no magnetic force acting on the ion these periods are equal, but though under the action of a magnetic force they are separated, their number cannot be increased. When therefore we get four or more lines, the inference is that the system giving the lines must have at least four degrees of freedom, and therefore must consist of more than one ion. The theory of a system of ions mutually influencing each other shows, as we should expect, that the effects are more complex than in the case of a single ion, and that the change in the frequency is not necessarily the same for all systems (see J. J. Thomson, Proc. Camb. Phil. Soc. 13, p. 39). Preston$^{49}$ and Runge and Paschen have proved that, in some cases at any rate, the change in the frequency of the different lines is of such a character that they can be grouped into series such that each line in the series has the same change in frequency for the same magnetic force, and, moreover, that homologous lines in the spectra of different metals belonging to the same group have the same change in frequency.
A very remarkable case of the Zeeman effect has been discovered by H. Becquerel and Deslandres (Comptes rendus, 127, p. 18). They found lines in iron when the most deflected components are those polarized in the plane at right angles to the magnetic force. On the simple theory the light polarized in this way is not affected. Thus the behaviour of the spectrum in the magnetic field promises to throw great light on the nature of radiation, and perhaps on the constitution of the elements. The study of these effects has been greatly facilitated by the invention by Michelson$^{50}$ of the echelon spectroscope.
There are some interesting phenomena connected with the Zeeman effect which are more easily observed than the effect itself. Thus Cotton$^{51}$ found that if we have two Bunsen flames, A and B, coloured by the same salt, the absorption of the light of one by the other is diminished if either is placed between the poles of a magnet: this is at once explained by the Zeeman effect, for the times of vibration of the molecules of the flame in the magnetic field are not the same as those of the other flame, and thus the absorption is diminished. Similar considerations explain the phenomenon observed by Egoroff and Georgiewsky,$^{52}$ that the light emitted from a flame in a transverse field is partially polarized in a plane parallel to the magnetic force; and also Righi’s$^{53}$ observation that if a sodium flame is placed in a longitudinal field between two crossed Nicols, and a ray of white light sent through one of the Nicols, then through the flame, and then through the second Nicol, the amount of light passing through the second Nicol is greater when the field is on than when it is off. Voight and Wiechert (Wied. Ann. 67, p. 345) detected the double refraction produced when light travels through a substance exposed to a magnetic field at right angles to the path of the light; this result had been predicted by Voight from theoretical considerations. Jean Becquerel has made some very interesting experiments on the effect of a magnetic field on the fine absorption bands produced by xenotime, a phosphate of yttrium and erbium, and tysonite, a fluoride of cerium, lanthanum and didymium, and has obtained effects which he ascribes to the presence of positive electrons. A very complete account of magneto- and electro-optics is contained in Voight’s Magneto- and Elektro-optik.
$^1$ Experimental Researches, Series 19. $^2$ Comptes rendus, 88, p. 709. $^3$ Wied. Ann. 6, p. 332; 8, p. 278; 10, p. 257. $^4$ Wied. Ann. 23, p. 228; 27, p. 191. $^5$ Wied. Ann. 31, p. 941. $^6$ Phil. Trans., A. 1885, Pt. 11, p. 343. $^7$ Wied. Ann. 26, p. 456. $^8$ Phil. Trans., A. 1895, Pt. 17, p. 621. $^9$ Wied. Ann. 24, p. 161. $^{10}$ Wied. Ann. 31, p. 970. 11 Comptes rendus, 57, p. 670. $^{12}$ Comptes rendus, 43, p. 529; 44, p. 1209. $^{13}$ Journ. Chem. Soc. 1884, p. 421; 1886, p. 177; 1887, pp. 362 and 808; 1888, p. 561; 1889, pp. 680 and 750; 1891, p. 981; 1892, p. 800; 1893, pp. 75, 99 and 488. 14 Wied. Ann. 44, p. 377. $^{15}$ Wied. Ann. 43, p. 280. $^{16}$ Zeitschrift f. physikal. Chem. 11, p. 753. $^{17}$ Phil. Mag. [5] 3, p. 321. $^{18}$ Ann. de chim. et de phys. [6] 4, p. 433; 9, p. 65; 10, p. 200. $^{19}$ Wied. Ann. 23, p. 228; 27, p. 191. $^{20}$ Wied. Ann. 39, p. 25. $^{21}$ Wied. Ann. 42, p. 115. $^{22}$ Phil. Mag. [5] 12, p. 171. $^{23}$ Journ. de Phys. 1884, p. 360. $^{24}$ Beiblätter zu Wied. Ann. 1885, p. 275. $^{25}$ Messungen über d. Kerr’sche Erscheinung. Inaugural Dissert. Leiden, 1893. $^{26}$ Phil. Mag. [5] 5, p. 161. $^{27}$ Phil. Mag. [3] 28, p. 469. $^{28}$ Die magn. Drehung d. Polarisationsebene des Lichts, Halle, 1863. $^{29}$ Electricity and Magnetism, chap. xxi. $^{30}$ Phil. Trans. 1880 (2), p. 691. $^{31}$ Phil. Mag. (5) 11, p. 254, 1881. $^{32}$ Arch. Néerl. 19, p. 123. $^{33}$ Wied. Ann. 23, p. 493; 67, p. 345. $^{34}$ Wied. Ann. 24, p. 119. $^{35}$ Wied. Beiblätter, 8, p. 869. $^{36}$ Comptes rendus, 108, p. 510. $^{37}$ Phil. Trans. 182, A. p. 371, 1892; Physical Optics, p. 393. $^{38}$ Wied. Ann. 46, p. 71; 47, p. 345; 48, p. 740; 50, p. 722. $^{39}$ Wied. Ann. 46, p.353; 48, p. 122; 49, p. 690. $^{40}$ Recent Researches, p. 489 et seq. $^{41}$ Phil. Trans., A. 1897, p. 89. $^{42}$ Brit. Assoc. Report, 1893. $^{43}$ Comptes rendus, 127, p. 548. $^{44}$ Bull. de l’Acad. des Sciences Belg. (3) 9, pp. 327, 381, 1885; 12 p. 30, 1886. $^{45}$ Communications from the Physical Laboratory, Leiden, No. 33, 1896; Phil. Mag. 43, p. 226; 44, pp. 55 and 255; and 45, p. 197. $^{46}$ Arch. Néerl. 25, p. 190. $^{47}$ Phil. Mag. 45, p. 325; 47, p. 165. $^{48}$ Comptes rendus, 126, p. 181. $^{49}$ Phil. Mag. 46, p. 187. $^{50}$ Phil. Mag. 45, p. 348. $^{51}$ Comptes rendus, 125, p. 865. $^{52}$ Comptes rendus, pp. 748 and 949, 1897. $^{53}$ Comptes rendus, 127, p. 216; 128, p. 45. (J. J. T.)