Page:CunninghamExtension.djvu/8

84 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.,

{{MathForm2|(8)|$$\left.\begin{array}{l} P_{X}=p_{x}\\ \\P_{Y}=\beta\left(1-\frac{vw_{x}}{c^{2}}\right)p_{y}+\frac{\beta vw_{y}}{c^{2}}p_{x}+\frac{\beta v}{c^{2}}m_{z}\\ \\P_{Z}=\beta\left(1-\frac{vw_{x}}{c^{2}}\right)p_{z}+\frac{\beta vw_{z}}{c^{2}}p_{x}-\frac{\beta v}{c^{2}}m_{y}\end{array}\right\} ,$$}}

{{MathForm2|(9)|$$\left.\begin{array}{l} M_{X}=m_{x}-\beta v\left(W_{Y}\delta m_{y}+W_{Z}\delta m_{z}\right)/c^{2}\\ \\M_{N}=\beta\left(1+\frac{vw_{x}}{c^{2}}\right)m_{n}\end{array}\right\} .$$}}

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.

The constitutive equations now become

{{MathForm2|(10)|$$\left.\begin{array}{rl} D_{X}= & \epsilon E_{X}\\ \\\left(1-\frac{v^{2}}{c^{2}}\right)D_{N}= & \left(\epsilon-\frac{v^{2}}{c^{2}}\right)E_{N}+\frac{1}{c}[v,\ \epsilon B-H]_{N}\end{array}\right\} ,$$}}

{{MathForm2|(12)|$$\left.\begin{array}{l} J_{X}=\sigma E_{X}/\beta\\ J_{N}=\beta\sigma\left(E_{n}+[v,B]_{n}/c\right)\end{array}\right\} .$$}}

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.