Page:The principle of relativity (1920).djvu/125

 Now upon the basis of the equations (55) and (56), and referring back to the expression (82) for L, and from 57) we obtain the following expressions as components of N,—

(92) N_{h} = - 1/2 ΦΦ̄[part]ε/[part]x_{h} - 1/2 ΨΨ̄[part]μ/[part]x_{h} + (εμ - 1)(Ω_{1}[part]ω_{1}/[part]x_{h} + Ω_{2}[part]ω_{2}/[part]x_{h} + Ω_{3}[part]ω_{3}/[part]x_{h} + Ω_{4}[part]ω_{4}/[part]x_{h})

for h = 1, 2, 3, 4.

Now if we make use of (59), and denote the space-vector which has Ω_{1}, Ω_{2}, Ω_{3} as the x, y, z components by the symbol W, then the third component of 92) can be expressed in the form

(93) (εμ - 1)/[sqrt](1 - u^2) (W[part]u/[part]x_{h}),

The round bracket denoting the scalar product of the vectors within it.

§ 14.

Let us now write out the relation K = lor S = -sF + N in a more practical form; we have the four equations

(94) K_{1} = [part]X_{x}/[part]x + [part]X_{y}/[part]y + [part]X_{y}/[part]z - [part]X_{t}/[part]t = ρE_{x} + s_{y}M_{z} - s_{z}M_{x}

- 1/2 ΦΦ̄[part]ε/[part]x - 1/2 ΨΨ̄[part]μ/[part]x + (εμ - 1)/[sqrt](1 - u^2) (W[part]u/[part]x),

(95) K_{2} = [part]Y_{x}/[part]x + [part]Y_{y}/[part]y + [part]Y_{z}/[part]z - [part]Y_{t}/[part]t = ρE_{y} + s_{z}M_{x} - s_{x}M_{y}

- 1/2 ΦΦ̄[part]ε/[part]y - 1/2 ΨΨ̄[part]μ/[part]y + (εμ - 1)/[sqrt](1 - u^2) (W[part]u/[part]y),