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of time to the application of the Lorentz-transformation. The paper of Einstein which has been cited in the Introduction, has succeeded to some extent in presenting the nature of the transformation from the physical standpoint.

PART II. ELECTRO-MAGNETIC PHENOMENA.

§ 7. Fundamental Equations for bodies at rest.

After these preparatory works, which have been first developed on account of the small amount of mathematics involved in the limitting case $\epsilon =1,\ \mu =1,\ \sigma =1$, let us turn to the electro-magnetic phenomena in matter. We look for those relations which make it possible for us — when proper fundamental data are given — to obtain the following quantities at every place and time, and therefore at every spacetime point as functions of x, y, z, t: — the vector of the electric force ${\mathfrak {E}}$, the magnetic induction ${\mathfrak {M}}$, the electrical induction ${\mathfrak {e}}$, the magnetic force ${\mathfrak {m}}$, the electrical space-density $\varrho$, the electric current ${\mathfrak {s}}$ (whose relation hereafter to the conduction current is known by the manner in which conductivity occurs in the process), and lastly the vector ${\mathfrak {w}}$, the velocity of matter.

The relations in question can be divided into two classes.

Firstly — those equations, which, — when ${\mathfrak {w}}$, the velocity of matter is given as a function of x, y, z, t, — lead us to a knowledge of other magnitude as functions of x, y, z, t — I shall call this first class of equations the fundamental equations —

Secondly, the expressions for the ponderomotive force, which, by the application of the Laws of Mechanics, gives us further information about the vector ${\mathfrak {w}}$ as functions of x, y, z, t.

For the case of bodies at rest, i.e. when ${\mathfrak {w}}(x,\ y,\ z,\ t)=0$, the theories of Maxwell (Heaviside, Hertz) and Lorentz lead to the same fundamental equations. They are ; —

(1) The Differential Equations: — which contain no constant referring to matter: —

${\begin{array}{rcrl}(I)&\qquad &curl\ {\mathfrak {m}}-{\frac {\partial e}{\partial t}}&={\mathfrak {s}},\\\\(II)&&div\ {\mathfrak {e}}&=\varrho ,\\\\(III)&&curl\ {\mathfrak {E}}+{\frac {\partial {\mathfrak {M}}}{\partial t}}&=0,\\\\(IV)&&div\ {\mathfrak {M}}&=0\end{array}}$;