Page:CunninghamExtension.djvu/8

and, similarly,

$\delta M_{N}=\beta \left(1+{\frac {vw_{x}}{c^{2}}}\right)\delta m_{n}.$

From these, by addition, the formulae of transformation are seen to be exactly those given by Mirimanoff, viz.,

$\left.{\begin{array}{l}P_{X}=p_{x}\\\\P_{Y}=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)p_{y}+{\frac {\beta vw_{y}}{c^{2}}}p_{x}+{\frac {\beta v}{c^{2}}}m_{z}\\\\P_{Z}=\beta \left(1-{\frac {vw_{x}}{c^{2}}}\right)p_{z}+{\frac {\beta vw_{z}}{c^{2}}}p_{x}-{\frac {\beta v}{c^{2}}}m_{y}\end{array}}\right\},$

(8)

$\left.{\begin{array}{l}M_{X}=m_{x}-\beta v\left(W_{Y}\delta m_{y}+W_{Z}\delta m_{z}\right)/c^{2}\\\\M_{N}=\beta \left(1+{\frac {vw_{x}}{c^{2}}}\right)m_{n}\end{array}}\right\}.$

(9)

It is thus verified, by direct method, that the equations of the field as given by Lorentz for moving ponderable bodies are left unchanged by a transformation in which the pairs of vectors $(e,b),\ (d,h-[pw]/c)$ are correlated in the same manner as were (e, h) for the free aether.^[1]

The constitutive equations now become

$\left.{\begin{array}{rl}D_{X}=&\epsilon E_{X}\\\\\left(1-{\frac {v^{2}}{c^{2}}}\right)D_{N}=&\left(\epsilon -{\frac {v^{2}}{c^{2}}}\right)E_{N}+{\frac {1}{c}}[v,\ \epsilon B-H]_{N}\end{array}}\right\},$

(10)

$\mu H=B+(\mu -1)[v,E]/c,$

(11)

$\left.{\begin{array}{l}J_{X}=\sigma E_{X}/\beta \\J_{N}=\beta \sigma \left(E_{n}+[v,B]_{n}/c\right)\end{array}}\right\}.$

(12)

The first and last of these appear to be the exact forms of the better known approximate equations

${\begin{array}{l}D=\epsilon E+(\epsilon -1)[w,B]/c,\\J=\sigma (E+[w,B]),\end{array}}$

and reduce to these, if $\mu =1$ and if $v^{2}/c^{2}$ is neglected.

The middle equation has as yet no experimental corroboration.

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1. Cf. Mirimanoff, loc. cit.