(M = magnetic induction, E = Electric force,
_f_ the space-time vector of the second kind (_m_,-_ie_)
(_m_ = magnetic force, _e_ = Electric Induction.
_s_ the space-time vector of the first kind (C, _i_ρ)
(ρ = electrical space-density, C - ρ_u_ = conductivity current,
ε = dielectric constant, μ = magnetic permeability,
σ = conductivity,
then the fundamental equations for electromagnetic processes in moving
bodies are[26]
{A} lor _f_ = -_s_
{B} log F* = 0
{C} ω_f_ = εωF
{D} ωF* = μω_f_*
{E} _s_ + (ω_ṡ_), _w_ = - σωF.
ω ῶ = -1, and ωF, ω_f_, ωF*, ω_f_*, _s_ + (ω_s_)ω which are space-time
vectors of the first kind are all normal to ω, and for the system {B},
we have
lor (lor F*) = 0.
Bearing in mind this last relation, we see that we have as many
independent equations at our disposal as are necessary for determining
the motion of matter as well as the vector _u_ as a function of _x_,
_y_, _z_, _t_, when proper fundamental data are given.
§ 13. The Product of the Field-vectors _f_ F.
Finally let us enquire about the laws which lead to the determination of
the vector ω as a function of (_x_, _y_, _z_, _t_.) In these
investigations, the expressions which are obtained by the multiplication
of two alternating matrices
_f_ = | 0 _f₁₂_ _f₁₃_ _f₁₄_ |
| _f₂₁_ 0 _f₂₃_ _f₂₄_ |
| _f₃₁_ _f₃₂_ 0 _f₃₄_ |
| _f₄₁_ _f₄₂_ _f₄₃_ 0 |
F = | 0 F₁₂ F₁₃ F₁₄ |
| F₂₁ 0 F₂₃ F₂₄ |
| F₃₁ F₃₂ 0 F₃₄ |
| F₄₁ F₄₂ F₄₃ 0 |
are of much importance. Let us write,
(70) _f_F =| S₁₁ - L S₁₂ S₁₃ S₁₄ |
| S₂₁ S₂₂ - L S₂₃ S₂₄ |
| S₃₁ S₃₂ S₃₃ - L S₃₄ |
| S₄₁ S₄₂ S₄₃ S₄₄ - L |
Then (71) S₁₁ + S₂₂ + S₃₃ + S₄₄ = 0.
Let L now denote the symmetrical combination of the indices 1, 2, 3, 4,
given by
(72) L = ½(_f₂₃_ F₂₃ + _f₃₁_F₃₁ + _f₁₂_ + F₁₂ + _f₁₄_ F₁₄
+ _f₂₄_ F₂₄ + _f₃₄_ F₃₄)