Page:Electromagnetic phenomena.djvu/15

 these symbols indicate the moment, the coordinates and the true time, whereas their meaning is different for the moving system, $$\mathfrak{p}'$$, x', y', z', t'  being here related to the moment $$\mathfrak{p}$$, the coordinates x, y, z and the general time t in the manner expressed by (26), (4) and (5).

It has already been stated that the equation (27) applies to both systems. The vector $$\mathfrak{d}'$$ will therefore be the same in Σ'  and Σ, provided we always compare corresponding places and times. However, this vector has not the same meaning in the two cases. In Σ'  it represents the electric force, in Σ it is related to this force in the way expressed by (20). We may therefore conclude that the electric forces acting, in Σ and in Σ' , on corresponding particles at corresponding instants, bear to each other the relation determined by (21). In virtue of our assumption b, taken in connexion with the second hypothesis of § 8, the same relation will exist between the "elastic" forces; consequently, the formula (21) may also be regarded as indicating the relation between the total forces, acting on corresponding electrons, at corresponding instants.

It is clear that the state we have supposed to exist in the moving system will really be possible if, in Σ and Σ' , the products of the mass in and the acceleration of an electron are to each other in the same relation as the forces, i. e. if

Now, we have for the accelerations

as may be deduced from (4) and (5), and combining this with (32), we find for the masses

$$m(\Sigma)=(k^{3}l,\ kl,\ kl)m(\Sigma')$$.

If this is compared to (31), it appears that, whatever be the value of l, the condition is always satisfied, as regards the masses with which we have to reckon when we consider vibrations perpendicular to the translation. The only condition we have to impose on l is therefore

$$\frac{d(klw)}{dw}=k^{3}l$$.

But, on account of (3),

$$\frac{d(kw)}{dw}=k^{3}$$,

so that we must put