Page:Grundgleichungen (Minkowski).djvu/18

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

§ 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 (Heaviside, ) and  lead to the same fundamental equations. They are ; —

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