Page:The principle of relativity (1920).djvu/117

 So the system of differential equations (A) can be expressed in the concise form

{A} lor f = -s,

and the system (B) can be expressed in the form

{B} log F* = 0.

Referring back to the definition (67) for log [=s], we find that the combinations lor ((lor f)), and lor ((lor F*) vanish identically, when [function] and F* are alternating matrices. Accordingly it follows out of {A}, that

(68) ([part]s_{1}/[part]x_{1}) + ([part]s_{2}/[part]x_{2}) + ([part]s_{3}/[part]x_{3}) + ([part]s_{4}/[part]x_{4}) = 0,

while the relation

(69) lor (lor F*) = 0, signifies that of the four equations in {B}, only three represent independent conditions.

I shall now collect the results.

Let ω denote the space-time vector of the first kind

(u/[sqrt](1 - u^{2}), i/[sqrt](1 - u^{2})) this line and next]

(u = velocity of matter),

F the space-time vector of the second kind (M,-iE)

(M = magnetic induction, E = Electric force,

[function] the space-time vector af the second kind (m,-ie)

(m = magnetic force, e = Electric Induction.

s the space-time vector of the first kind (C, iρ)

(ρ = electrical space-density, C - ρu] = conductivity current,

ε = dielectric constant, μ = magnetic permeability,

σ = conductivity,