# Page:CunninghamExtension.djvu/15

1909.]
91
The principle of relativity in electrodynamics.

Hence

${\displaystyle {\frac {m\ ds}{M\ dS}}={\frac {n\ ds}{N\ dS}}=-\lambda ^{2}\beta ,}$

so that

${\displaystyle {\frac {l}{L}}={\frac {-m/\beta }{M}}={\frac {-n/\beta }{N}}={\frac {1}{\kappa }}}$ (say);

and therefore

${\displaystyle {\frac {F_{R}}{L}}={\frac {F_{\theta }}{M}}={\frac {F_{\phi }}{N}}.}$

Again,

${\displaystyle LH_{r}+MH_{\theta }+NH_{\phi }=-\lambda ^{2}\kappa \left(lh_{r}+mh_{\theta }+nh_{\phi }\right),}$

since

${\displaystyle me_{\phi }-ne_{\phi }=0.}$

Hence

${\displaystyle LH_{r}+MH_{\theta }+NH_{\phi }=0.}$

Thus the conditions for a moving reflecting surface are satisfied at the transformed surface.

Thus a perfect reflector transforms into a perfect reflector.

6. There is no difficulty in adapting the analysis of § 2 to the new transformation if the following form of it is noticed.

The differential elements in the transformation are connected by the equation

 ${\displaystyle \delta R=\lambda ^{-1}\beta (-\delta r+v\delta t),}$ ${\displaystyle \delta T=\lambda ^{-1}\beta \left(\delta t-{\frac {v}{c^{2}}}\delta r\right),}$

where, as before,

${\displaystyle \lambda =\left(r^{2}-c^{2}t^{2}\right)/k^{2},\ v=2c^{2}rt\left(r^{2}+c^{2}t^{2}\right),\ \beta =\left(1-v^{2}/c^{2}\right)^{-{\frac {1}{2}}}.}$

Thus within a small element of volume the space time coordinates are changed by a transformation of the same form as the Lorentz-Einstein, save for the magnification factor ${\displaystyle \lambda ^{-1}}$, and a difference of sign, which itself disappears in a sequence of two such transformations, e.g., in an infinitesimal transformation of the group.

The same is true of the fundamental equations connecting the magnetic and electric intensities in the two systems.

If, therefore, we confine our attention to a portion of matter contained within a volume which is small, but large enough to allow of the process of averaging commonly employed in molecular physics, we shall obtain results similar to those of the last section. It will be enough here to give the equations without the analysis, referring to the corresponding equations of previous sections.