the co-ordinates of an ion, R1 the coefficient of resistance to the motion of the ions, and a. the force at unit distance tending to bring the ion back to its position of equilibrium, K0 the specific inductive capacity of a vacuum. If the variables are proportional to el('" '”) we find by substitution that q is given by the equation n 2 zp SHQ3
Q2 " K0P2"P£i1i. ilizpz = ipezpz
where P = (a — § 1rne') -f- Rap -mpf,
or, by neglecting R, P=m(s7-p'), where s is the period of the free ions. If, q§ , qi are the roots of this equation, then corresponding to gl we have X0=LY0 and to gg X0= — LYQ. We thus get two oppositely circular-polarized rays travelling with the velocities p/ql and p/qz respectively. Hence if vi, 112 are these velocities, and v the velocity when there is no magnetic field, we obtain, if we neglect terms in HY,
v12 WZ 1nZ(S2 p2)2
U22 -92 m2(Y2 P2)2
The rotation r of the plane of polarization per unit length
- 1 (L L> 21rne3Hp2'v
“EP 2,1 U, 'm2(Sz p'2')z
Since I/v2=Ko-l-41rne'/m(s2-p'), we have if it is the refractive index for light of frequency p, and 'UD the velocity of light in vacuo.
- r2- 1 =41rne'v'0/m(s' -p7) ...... (I)
So that we may put
r = (ii - I>2P2I'l/511111167103 ..... (2) Becquerel (Comptes rendus, 125, p. 683) gives for r the expression L Ii Q
E-m 7J0 dh
where A is the wave length. This is equivalent to (2) if, u 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 19 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 Corbinof”
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 Ae7<P7'4'), if the reflected beam in which the electric force is parallel to x is represented by Be'f1"+'1'> and the reflected beam in which the electric force is parallel to the axis of y by Ce'<P'+'1'), 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 YD, X0 in the medium, give A = B = LC
(q+qi> (q-tai) (92-sziqz) q<q2-Qi)
or approximately, since qi and gg are nearly equal, L§ =q(q2-qi) =(/1”-KWH,
B q'-qi* 41f/HWVU
Thus in transparent bodies for which p is real, C and B differ in phase by -rr/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 p. contains an imaginary term, C/B will contain a real part so that the reflected light will be elliptically polarized, but the mailor axis is no longer in the plane of polarization of the incident lig t; we should thus have a rotation of the plane of polarization superposed on the elliptic polarization.
Zeeman's Eject.~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, ““ a Belgian physicist, noticed that the spectrum of a sodium flame was changed slightly in appearance by a magnetic held; 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." 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 mfg-l-ax= -Hag?
d* d .
ma;-I-az = 0.
The solution of these equations is
x = Acos (pit-I-(3)-I-B cos (pzt-I-Bl)
y 2 A Slfl -B Sill
z = C cos (pt-l-fy)
where a - mpi2= ~Hep1
G. -" WIP22 = HCP2
P' = 11/M.
or approximately pl =p-I-éii, p2 =p - é-7, -5° 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 pl and pg 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 pl and P2 giving the circularly polarized constituents of the doublet. Now suppose the magnetic force is at right angles to the direction of propa ation 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 w ve-front of the circular orbits at right angles to the magnetic force will be rectilinear motions of frequency pl and P2 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 107. 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 ro7 to 3'4 X 107, the value of e/m for the particle in the cathode rays is 1~7~ X IO7. The majority of the determinations of e/m from the Zeeman effect give numbers larger than this, the maximum being about twice this value.