Page:The principle of relativity (1920).djvu/80



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 limitting case ε = 1, μ = 1, σ = 0, they should constitute the laws for ponderable bodies. In this ideal limitting 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

(i) Curl m - (δe/δt) = ρu

(ii) div e = ρ

(iii) Curl e + δm/δt = 0

(iv) div m = 0

I shall now write (x_{1} x_{2} x_{3} x_{4}) for (x, y, z, t) and (ρ_{1}, ρ_{2}, ρ_{3}, ρ_{4}) for

(ρu_{x}, ρu_{y}, ρu_{z}, iρ)

i.e. the components of the convection current ρu, and the electric density multiplied by [sqrt]-1

Further I shall write

[function]_{2 3}, [function]_{3 1}, [function]_{1 2}, [function]_{1 4}, [function]_{2 4}, [function]_{3 4}.

for

m_{x}, m_{y}, m_{z}, -ie_{x}, -ie, -ie_{z}.

i.e., the components of m and (-i.e.) along the three axes; now if we take any two indices (h. k) out of the series

3, 4), [function]_{k h} = -[function]_{k h},