Page:The principle of relativity (1920).djvu/122

 right side of (83) as well as S_{k h} represent a space time vector of the second kind. Remembering this fact, it suffices, for establishing the theorems (82) and (83) generally, to prove it for the special case ω_{1} = 0, ω_{2} = 0, ω_{3} = 0, ω_{4} = i. But for this case ω = 0, we immediately arrive at the equations (82) and (83) by means (45), (51), (60) on the one hand, and e = εE, M = μm on the other hand.

The expression on the right-hand side of (81), which equals

[1/2 (m M - eE)^2] + (em) (EM),

is [>=] 0, because (em = ε Φ Ψ̄, (EM) = μ Φ Ψ̄; now referring back to 79), we can denote the positive square root of this expression as Det^{1/4} S.

Since [=[function]] = -[function], and F̄ = -F, we obtain for S̄, the transposed matrix of S, the following relations from (78),

(84) F[function] = S̄ - L, [function]^* F^* = -S̄ - L,

Then is S̄ - S = | S_{h k} - S_{t k} |

an alternating matrix, and denotes a space-time vector of the second kind. From the expressions (83), we obtain,

(85) S - S̄ = - (εμ - 1) [ω, Ω],

from which we deduce that [see (57), (58)].

(86) ω (S - S̄)^* = 0,

(87) ω (S - S̄) = (εμ - 1) Ω

When the matter is at rest at a space-time point, ω = 0, then the equation 86) denotes the existence of the following equations

Z_{y} = Y_{z}, X_{z} = Z_{x}, Y_{x} = X_{y},