Page:The principle of relativity (1920).djvu/100

 and M´ shall stand to E + [uM,], and M - [uE] in the same relation us e´ and m´ to e + [um], and m - (ue). From the relation e´ = εE´, the following equations follow

(C) e + [um] = ε(E + [uM]),

and from the relation M´ = μm´, we have

(D) M - [uE] = μ(m - [ue]),

For the components in the directions perpendicular to u, and to each other, the equations are to be multiplied by [sqrt](1 - u^2).

Then the following equations follow from the transfermation equations (12), (10), (11) in § 4, when we replace q, r_{v}, r_{[=v]}, t, r´_{v}, r´_{[=v]}, t' by |u|, C_{u}, C_{[=u]}, ρ, C´_{u}, C´_{[=u]}, ρ´

ρ´ = (-|u|C_{u} + ρ)/[sqrt](1 - u^2), C'_{u} = (C_{u} - |u|ρ)/[sqrt](1 - u^2), C´_{[=u]} = C_{=u},

E) (C_{u} - |u|ρ)/[sqrt](1 - u^2) = σ(E + [uM])_{u},

C_{[=u]} = σ (E + [uM])_{u}/[sqrt](1 - u^2)

In consideration of the manner in which σ enters into these relations, it will be convenient to call the vector C - ρu with the components C_{u} - ρ|u| in the direction of u, and C´_{=u} in the directions [=u] perpendicular to u the "Convection current." This last vanishes for σ = 0.

We remark that for ε = 1, μ = 1 the equations e´ = E´, m´ = M´ immediately lead to the equations e = E, m = M by means of a reciprocal Lorentz-transformation with -u as vector; and for σ = 0, the equation C´ = 0 leads to C = ρu; that the fundamental equations of Äther discussed in § 2 becomes in fact the limitting case of the equations obtained here with ε = 1, μ = 1, σ = 0.