Page:The principle of relativity (1920).djvu/54

 which are defined by their pondermotive reaction, the same equations hold, i.e.

(1/c) [part]/[part]τ (X´, Y´, Z´)] = | [part]/[part]ξ [part]/[part]η [part]/[part]ζ |
 * L´ M´ N´ |,

(1/c) [part]/[part]τ (L´, M´, N´) = | [part]/[part]ξ [part]/[part]η [part]/[part]ζ |
 * X´ Y´ Z´ | (3)

Clearly both the systems of equations (2) and (3) developed for the system k shall express the same things, for both of these systems are equivalent to the Maxwell-Hertzian equations for the system K. Since both the systems of equations (2) and (3) agree up to the symbols representing the vectors, it follows that the functions occurring at corresponding places will agree up to a certain factor ψ(v), which depends only on v, and is independent of (ξ, η, ζ, τ). Hence the relations,

[X´, Y´, Z´] = ψ(v) [X, β(Y - (v/c)N), β(Z + (v/c)M)],

[L´, M´, N´] = ψ(v) [L, β(M + (v/c)Z), β(N - (v/c)Y)].

Then by reasoning similar to that followed in §(3), it can be shown that ψ(v) = 1.

[X´, Y´, Z´] = [X, β(Y - (v/c)N), β(Z + (v/c)M)]

[L´, M´, N´] = [L, β(M + (v/c)Z), β(N - (v/c)Y)].