Page:CunninghamExtension.djvu/2

so that e, h are the electric and magnetic intensities, ρ is the density of electricity, and u the velocity of convection in a sequence of electromagnetic phenomena in the æther, the observer being at rest relative to the æther, then the quantities E, H, U, Ρ given by the equations

${\begin{array}{lllll}E_{X}=e_{x},&&E_{Y}=\beta \left(e_{y}-vh_{z}/c\right),&&E_{Z}=\beta \left(e_{z}+vh_{y}/c\right);\\H_{X}=h_{x},&&H_{Y}=\beta \left(h_{y}+ve_{z}/c\right),&&H_{Z}=\beta \left(h_{z}-ve_{y}/c\right);\end{array}}$ ${\begin{array}{l}{\mathsf {\mathrm {P} }}=\beta \rho \left(1-vu_{x}/c^{2}\right),\\U=B^{-1}\left(1-vu_{x}/c^{2}\right)^{-1}\left\{\beta \left(u_{x}-v\right),\ u_{y},\ u_{z}\right\},\end{array}}$

of which the last is deduced without further assumption from (1), satisfy the same equations in (X, Y, Z, T).

Thus E, H, U, Ρ are quantities which can consistently represent a sequence of phenomena in which the second observer is supposed to be at rest in the æther.

The relations given above are reciprocal, and it is proved that the charges in corresponding elements of volume at corresponding times are the same in magnitude. Thus it is impossible for either observer to deduce from any experimental observation that he is at rest in the true æther and that the other is moving or vice versa.

This being so, the relations connecting the values, as estimated by the two observers of all physical quantities, must be such as to leave the constitutive equations for material media invariant also. This aspect of the question has been discussed by Minkowski,^[1] Frank,^[2] and Mirimanoff.^[3] The two former agree in stating that the Lorentz equations must be modified if the theorem of relativity is to be maintained in its entirety; but the last-named shews that this may be avoided, and that the relativity is complete and exact provided a change is made in the constitutive equations as ordinarily given. The modification required is of the second order of small quantities only in the equation

$D=\epsilon E+(\epsilon -1)[U,B]/c,$

which has been verified to the first order by H. A. Wilson, and is of the first order in u/c in the equation $B=\mu H$. Thus the suggested equations are not contrary to experiment. The first part of the present paper is a verification of Mirimanoff's results by a process of averaging.^[4]

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1. Gött. Nachr., 1908, p. 53.
2. Ann. der Phys., 27, 1908, p. 1059.
3. Ann. der Phys., 28, 1909, p. 192.
4. Frank (Ann. der Phys., 27, 1908) gives a verification of Minkowski's result by a similar process, but the investigation does not appear to take into account the Röntgen current-curl [pu].