The formula which expresses the relations is one from Lorenz ("Annalen der Physik und Chemie," vii, p. 161):
where T = time of oscillation of the electrical wave, P = the self-induction of the conductor concerned, C = its electrostatic capacity, and A = velocity of electrical propagation, which is assumed to be that of light. It will thus be seen that each conductor has its own proper time of electrical oscillation and wave-length.
If, now, the capacity of one side of the rectangle be increased, the time of oscillation of the waves on that side will be also increased. This will increase the wave-length, and equilibrium can be established by adding the same capacity to the other side, or by changing the point of contact.
For the reason that the only variables in the time of oscillation are the self-induction and the capacity, the resistance and material of the rectangle have no influence on the phenomena. Because the capacity of each half of the rectangle is chiefly that of the balls at its terminals, the employing of fine wire for one half can produce no noticeable effect.
That the size of the rectangle should have such an influence is to be expected up to certain limits—that is, until the total length of the sides is one wave-length or a multiple of the same. Then the waves could be made to arrive at the terminals in opposite phases, and would give the largest sparks.
Were this the only proof which Hertz could give of interference, a great deal of doubt might be cast upon its conclusiveness. Would not one naturally expect that, if both sides of the rectangle were of the same length and had the same capacity, the potential on both balls would be the same, and no discharge could take place; or, when of different capacities, the charging and discharging following each other so rapidly that the same quantity of electricity would tend to pass through a section of each side of the rectangle, and would thus necessitate a discharge?
But Hertz's quantitative experiments are more satisfactory. In order to understand them, a few preliminary phenomena must be described. These relate to what he calls the principle of "resonance." As any sound resonator, having its own proper wave-length, can be set in vibration by a vibrating body of the same or multiple time of vibration, so we might suppose that any electrical conductor could be set in vibration by a neighboring electrical wave disturbance of proper time of oscillation. This supposition is verified by experiment.
The apparatus and arrangement are very similar to those in the previous experiment. However, instead of the two outer brass balls on the Ruhmkorff discharger, two hollow zinc spheres of