Page:Electromagnetic phenomena.djvu/16

${\frac {dl}{dw}}=0,\ l=const$.

The value of the constant must be unity, because we know already that, for w=0, l=1.

We are therefore led to suppose that the influence of a translation on the dimensions (of the separate electrons and of a ponderable body as a whole) is confined to those that have the direction of the motion, these becoming k times smaller than they are in the state of rest. If this hypothesis is added to those we have already made, we may be sure that two states, the one in the moving system, the other in the same system while at rest, corresponding as stated above, may both be possible. Moreover, this correspondence is not limited to the electric moments of the particles. In corresponding points that are situated either in the aether between the particles, or in that surrounding the ponderable bodies, we shall find at corresponding times the same vector ${\mathfrak {d}}'$ and, as is easily shown, the same vector ${\mathfrak {h}}'$. We may sum up by saying : If, in the system without translation, there is a state of motion in which, at a definite place, the components of ${\mathfrak {p}}$, ${\mathfrak {d}}$, and ${\mathfrak {h}}$ are certain functions of the time, then the same system after it has been put in motion (and thereby deformed) can be the seat of a state of motion in which, at the corresponding place, the components of ${\mathfrak {p}}'$, ${\mathfrak {d}}'$, and ${\mathfrak {h}}'$ are the same functions of the local time.

There is one point which requires further consideration. The values of the masses m₁, and m₂ having been deduced from the theory of quasi-stationary motion, the question arises, whether we are justified in reckoning with them in the case of the rapid vibrations of light. Now it is found on closer examination that the motion of an electron may be treated as quasi-stationary if it changes very little during the time a light-wave takes to travel over a distance equal to the diameter. This condition is fulfilled in optical phenomena, because the diameter of an electron is extremely small in comparison with the wave-length.

§ 11. It is easily seen that the proposed theory can account for a large number of facts.

Let us take in the first place the case of a system without translation, in some parts of which we have continually ${\mathfrak {p}}=0$, ${\mathfrak {d}}=0$ and ${\mathfrak {h}}=0$. Then, in the corresponding state for the moving system, we shall have in corresponding parts (or, as we may say, in the same parts of the deformed system) ${\mathfrak {p}}'=0$, ${\mathfrak {d}}'=0$ and ${\mathfrak {h}}'=0$. These equations implying ${\mathfrak {p}}=0$, ${\mathfrak {d}}=0$, ${\mathfrak {h}}=0$, as is seen by (26) and (6), it appears