Page:The principle of relativity (1920).djvu/224

 From (59), it follows that the system of equations

(60) [part]F_{ρσ}/[part]x_{τ} + [part]F_{στ}/[part]x_{ρ} + [part]F_{τρ}/[part]x_{σ} = 0

is satisfied of which the left-hand side, according to (37), is an anti-symmetrical tensor of the third kind. This system (60) contains essentially four equations, which can be thus written:—

{ [part]F_{2 3}/[part]x_{4} + [part]F_{3 4}/[part]x_{2} [part]F_{4 2}/[part]x_{3} = 0 {     { [part]F_{3 4}/[part]x_{1} + [part]F_{4 1}/[part]x_{3} [part]F_{1 3}/[part]x_{4} = 0 (60a) { { [part]F_{4 1}/[part]x_{2} + [part]F_{1 2}/[part]x_{4} [part]F_{2 4}/[part]x_{1} = 0 {     { [part]F_{1 2}/[part]x_{3} + [part]F_{2 3}/[part]x_{1} [part]F_{3 1}/[part]x_{2} = 0.

This system of equations corresponds to the second system of equations of Maxwell. We see it at once if we put

{ F_{2 3} = H_{x} F_{1 4} = E_{x} { (61) { F_{3 1} = H_{y} F_{2 4} = E_{y}. {    { F_{1 2} = H_{z} F_{3 4} = E_{z}

Instead of (60a) we can therefore write according to the usual notation of three-dimensional vector-analysis:—

{ [part]H/[part]t + rot E = 0 (60b) { { div H = 0.