Page:The principle of relativity (1920).djvu/121

 (79) SS = L^2 - Det^{1/2}[function] Det^{1/2}F

i.e. the product of the matrix S into itself can be expressed as the multiple of a unit matrix—a matrix in which all the elements except those in the principal diagonal are zero, the elements in the principal diagonal are all equal and have the value given on the right-hand side of (79). Therefore the general relations

(80) S_{h1} S_{1k} + S_{h2} S_{2k} + S_{h3} S_{3k} + S_{h4} S_{4k} = 0,

h, k being unequal indices in the series 1, 2, 3, 4, and

(81) S_{h1} S_{1h} + S_{h2} S_{2h} + S_{h3} S_{3h} + S{h4} S_{4h} = L^2 - Det^{1/2}[function] Det^{1/2}F,

for h = 1, 2, 3, 4.

Now if instead of F, and [function] in the combinations (72) and (73), we introduce the electrical rest-force Φ, the magnetic rest-force Ψ, and the rest-ray Ω [(55), (56) and (57)], we can pass over to the expressions,—

(82) L = - 1/2 ε Φ Φ̄ + 1/2 μ Ψ Ψ̄,

(83) S_{hk} = - 1/2 ε Φ Φ̄ e_{hk} - 1/2 μ Ψ Ψ̄ e_{hk} + ε (Φ_{h} Φ_{k} - Φ (Φ̄) ω_{h} [omega_{k} + μ (Ψ_{h} Ψ_{k} - [Psi} Ψ̄ [omega_{h} ω_{k}) - Ω_{h} ω_{k} - εμ ω_{h} Ω_{k} (h_{1} k = 1, 2, 3, 4).

Here we have

ΦΦ̄ = Φ_{1}^2 + Φ_{2}^2 + Φ_{3}^2 + Φ_{4}^2, ΨΨ̄ = Ψ_{1}^2 + Ψ_{2}^2 + Ψ_{3}^2 + Ψ_{4}^2

e_{hh} = 1, e_{hk} = 0 (h [/=] k).

The right side of (82) as well as L is an invariant in a Lorentz transformation, and the 4 × 4 element on the