Page:Electromagnetic phenomena.djvu/3

to the existing theory of electric and optical phenomena in moving bodies that, in order to explain Michelsons's negative result, the introduction of a new hypothesis has been required, and that the same necessity may occur each time new facts will be brought to light. Surely, this course of inventing special hypothesis for each new experimental result is somewhat artificial. It would be more satisfactory, if it were possible to show, by means of certain fundamental assumptions, and without neglecting terms of one order of magnitude or another, that many electromagnetic actions are entirely independent of the motion of the system. Some years ago, I have already sought to frame a theory of this kind.^[1] I believe now to be able to treat the subject with a better result. The only restriction as regards the velocity will be that it be smaller than that of light.

§ 3. I shall start from the fundamental equations of the theory of electrons.^[2] Let ${\mathfrak {d}}$ be the dielectric displacement in the aether, ${\mathfrak {h}}$ the magnetic force, $\varrho$ the volume-density of the charge of an electron, ${\mathfrak {v}}$ the velocity of a point of such a particle, and ${\mathfrak {f}}$ the electric force, i.e. the force, reckoned per unit charge, which is exerted by the aether on a volume-element of an electron. Then, if we use a fixed system of coordinates,

${\begin{cases}div\ {\mathfrak {d}}=\varrho ,\quad div\ {\mathfrak {h}}=0,\\rot\ {\mathfrak {h}}={\frac {1}{c}}\left({\dot {\mathfrak {d}}}+\varrho {\mathfrak {v}}\right),\\rot\ {\mathfrak {d}}=-{\frac {1}{c}}{\dot {\mathfrak {h}}},\\{\mathfrak {f}}={\mathfrak {d}}+{\frac {1}{c}}\left[{\mathfrak {v.h}}\right].\end{cases}}$.

(2)

I shall now suppose that the system as a whole moves in the direction of x with a constant velocity w, and I shall denote by ${\mathfrak {u}}$ any velocity a point of an electron may have in addition to this, so that

${\mathfrak {v}}_{x}={\mathfrak {w}}+{\mathfrak {u}}_{x},\quad {\mathfrak {v}}_{y}={\mathfrak {u}}_{y},\quad {\mathfrak {v}}_{z}={\mathfrak {u}}_{z}$.

If the equations (2) are at the same time referred to axes moving with the system, they become

1. Lorentz, Zittingsverslag Akad. v. Wet., 7 (1899), p. 507, Amsterdam Proc., 1898-99, p. 427
2. M. E., § 2.