Let a rectangular system (_x_, _y_, _z_, _t_,) of reference be given in
space and time. The unit of time shall be chosen in such a manner with
reference to the unit of length that the velocity of light in space
becomes unity.
Although I would prefer not to change the notations used by Lorentz, it
appears important to me to use a different selection of symbols, for
thereby certain homogeneity will appear from the very beginning. I shall
denote the vector electric force by E, the magnetic induction by M, the
electric induction by _e_ and the magnetic force by _m_, so that (E, M,
_e_, _m_) are used instead of Lorentz’s (E, B, D, H) respectively.
I shall further make use of complex magnitudes in a way which is not yet
current in physical investigations, _i.e._, instead of operating with
(_t_), I shall operate with (_i t_), where _i_ denotes √(-1). If now
instead of (_x_, _y_, _z_, _i t_), I use the method of writing with
indices, certain essential circumstances will come into evidence; on
this will be based a general use of the suffixes (1, 2, 3, 4). The
advantage of this method will be, as I expressly emphasize here, that we
shall have to handle symbols which have apparently a purely real
appearance; we can however at any moment pass to real equations if it is
understood that of the symbols with indices, such ones as have the
suffix 4 only once, denote imaginary quantities, while those which have
not at all the suffix 4, or have it twice denote real quantities.
An individual system of values of (_x_, _y_, _z_, _t_) _i. e._, of (_x₁_
_x₂_ _x₃_ _x₄_) shall be called a space-time point.
Further let _u_ denote the velocity vector of matter, ε the dielectric
constant, μ the magnetic permeability, σ the conductivity of matter,
while ρ denotes the density of electricity in space, and _x_ the vector
of “Electric Current” which we shall some across in §7 and §8.
PART I
§ 2.
The Limiting Case.
The Fundamental Equations for Äther.
By using the electron theory, Lorentz in his above mentioned essay
traces the Laws of Electro-dynamics of Ponderable Bodies to still
simpler laws. Let us now adhere to these simpler laws, whereby we
require that for the limiting case ε = 1, μ = 1, σ = 0, they should
constitute the laws for ponderable bodies. In this ideal limiting case ε
= 1, μ = 1, σ = 0, E will be equal to _e_, and M to _m_. At every space
time point (_x_, _y_, _z_, _t_) we shall have the equations[15]
(i) Curl _m_ - (δ_e_/δ_t_) = ρu
(ii) div _e_ = ρ
(iii) Curl _e_ + δ_m_/δ_t_ = 0
(iv) div m = 0
I shall now write (_x₁_ _x₂_ _x₃_ _x₄_) for (_x_, _y_, _z_, _t_) and
(ρ₁, ρ₂, ρ₃, ρ₄) for
$$ (\rho u_{x}, \rho u_{y}, \rho u_{z}, i\rho) $$
_i.e._ the components of the convection current ρu, and the electric
density multiplied by √ -1