It has been indicated that the differential equation does not represent oscillatory motion if the value of R is too great. The exact condition depends on the roots of the quadratic equation Lx2 + Rx + 1 ⁄ C = 0, obtained by writing 1 for Q, and x for d ⁄ dt, and then treating x as a quantity. These roots are − R ⁄ 2L ± √(R2 ⁄ 4L2 − 1 ⁄ CL), and are therefore real or imaginary according as 4L ⁄ C is less or greater than R2. If the roots are real, that is, if R2 be greater than 4L ⁄ C, the discharge will not be oscillatory; the Faraday tubes referred to above will be absorbed in the wire without any return to the condenser. The corresponding result happens with the vibrator when R is sufficiently great, or L ⁄ C sufficiently small (a weak spring and a small mass, or both), to enable the condition to be fulfilled.
If, however, the roots of the quadratic are imaginary, that is, if 4L ⁄ C be greater than R2 (a condition which will be fulfilled in the spring analogue, by making the spring sufficiently stiff and the mass large enough to prevent the friction from controlling the motion) the motion is one in which Q disappears by oscillations about zero, of continually diminishing amplitude. A complete discussion gives for the period of oscillation 4πL ⁄ √(4L ⁄ C − R2), or if R be comparatively small, 2π√(LC). The charge Q falls off by the fraction e − RT⁄2L (where e is the number 2.71828...) in each period T, and so gradually disappears.
Thus electric oscillations are produced, that is to say, the charged state of the condenser subsides by oscillations, in which the charged state undergoes successive reversals, with dissipation of energy in the wire; and both the period and the rate of dissipation can be calculated if L, C, and R are known, or can be found, for the system. These quantities can be calculated and adjusted in certain definite cases, and as the electric oscillations can be experimentally observed, the theory can be verified. This has been done by various experimenters.
Returning to the pendulum illustration, it will be seen that the pendulum held deflected is analogous to the charged jar, letting the pendulum go corresponds to connecting the discharging coil to the coatings, the motion of the pendulum is the analogue of that motion of the medium in which consists the magnetic field, the friction of the air answers to the resistance of the wire which finally damps out the current. The inertia or mass of the bob is the analogue of what Thomson called the electromagnetic inertia of the coil and connections; what is now generally called the self-inductance of the conducting system. The component of gravity along the path towards the lowest point, answers to the reciprocal, 1 ⁄ C, of the capacity of the condenser.
It appears from the analogy that just as the oscillations of a pendulum can be prevented by immersing the bob in a more resisting medium, such as treacle or oil, so that when released the pendulum slips down to the vertical without passing it, so by properly proportioning the resistance in the circuit to the electromagnetic inertia of the coil, oscillatory discharge of the Leyden jar may also be rendered impossible.
All this was worked out in an exceedingly instructive manner in Thomson's paper; the account of the matter by the motion of Faraday tubes is more recent, and is valuable as suggesting how the inertia effect of the coil arises. The analogy of the pendulum is a true one, and enables the facts to be described; but it is to be remembered that it becomes evident only as a consequence of the mathematical treatment of the electrical problem. The paper was of great importance for the investigation of the electric waves used in wireless telegraphy in our own time. It enabled the period of oscillation of different systems to be calculated, and so the rates of exciters and receivers of electric waves to be found. For such vibrators are really Leyden jars, or condensers, caused to discharge in an oscillatory manner.
This application was not foreseen by Thomson, and, indeed, could hardly be, as the idea of electric waves in an insulating medium came a good deal later in the work of Maxwell. Yet the analogy of the pendulum, if it had then been examined, might have suggested such waves. As the bob oscillates backwards and forwards the air in which it is immersed is periodically disturbed, and waves radiate outwards from it through the surrounding atmosphere. The energy of these waves is exceedingly small, otherwise, as pointed out above, a term would have to be included in the theory of the resisted motion of the pendulum to account for this energy of radiation. So likewise when the electric vibrations proceed, and the insulating medium is the seat of a periodically varying magnetic field, electromagnetic waves are propagated outwards through the surrounding medium—the ether—and the energy carried away by the waves is derived from the initial energy of the charged condenser. In strictness also Thomson's theory of electric oscillations requires an addition to account for the energy lost by radiation. This is wanting, and the whole decay of the amount of energy present at the oscillator is put down to the action of resistance—that is, to something of the nature of frictional retardation. Notwithstanding this defect of the theory, which is after all not so serious as certain difficulties of exact calculation of the self-inductance of the discharging conductor, the periods of vibrators can be very accurately found. When these are known it is only necessary to measure the length of an electrical wave to find its velocity of propagation. When electromagnetic waves were discovered experimentally in 1888 by Heinrich Hertz, it was thus that he was able to demonstrate that they travelled with the velocity of light.
Thomson suggested that double, triple and quadruple flashes of lightning might be successive flashes of an oscillatory discharge. He also pointed out that if a spark-gap were included in a properly arranged condenser and discharging wire, it might be possible, by means of Wheatstone's revolving mirror, to see the sparks produced in the successive oscillations, as "points or short lines of light separated by dark intervals, instead of a single point of light, or of an unbroken line of light, as it would be if the discharge were instantaneous, or were continuous, or of appreciable duration."
This anticipation was verified by experiments made by Feddersen, and published in 1859 (Pogg. Ann., 108, 1859). The subject was also investigated in Helmholtz's laboratory at Berlin, by N. Schiller, who, determining the period for condensers with different substances between the plates, was able to deduce the inductive capacities of these substances (Pogg. Ann., 152, 1874). [The specific inductive capacity of an insulator is the ratio of the capacity of a condenser with the substance between the plates to the capacity of an exactly similar condenser with air between the plates.]
The particular case of non-oscillatory discharge obtained by supposing C and Q both infinitely great and to have a finite ratio V (which will be the potential, p. [34], of the charged plate), is considered in the paper. The discharging conductor is thus subjected to a difference of potential suddenly applied and maintained at one end, while the other end is kept at potential zero. The solution of the differential equation for this case will show how the current rises from zero in the wire to its final steady value. If c be put as before for the current − dQ ⁄ dt, and the constant value V for Q ⁄ C, the equation is