| K0 | dY0 | + 4πne | dη | = − | dα |
| dt | dt | dz |
| m | d²ξ | + R1 | dξ | + aξ = ( X0 + | 4π | neξ ) e + He | dη |
| dt² | dt | 3 | dt |
| m | d²η | + R1 | dη | + aη = ( Y0 + | 4π | neη ) e − He | dξ | ; |
| dt² | dt | 3 | dt |
where H is the external magnetic field, X0, Y0 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, R1 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, K0 the specific inductive capacity of a vacuum. If the variables are proportional to εl(pt−qz) we find by substitution that q is given by the equation
| q² − K0p² − | 4πne²p²P | = ± | 4πne³Hp³ | , |
| P² − H²e²p² | P² − H²e²p² |
where
P = (a − 4⁄3πne²) + R1ιp − mp²,
or, by neglecting R, P = m (s² − p²), where s is the period of the free ions. If, q1², q2² are the roots of this equation, then corresponding to q1 we have X0 = ιY0 and to q2 X0 = −ιY0. We thus get two oppositely circular-polarized rays travelling with the velocities p/q1 and p/q2 respectively. Hence if v1, v2 are these velocities, and v the velocity when there is no magnetic field, we obtain, if we neglect terms in H²,