Page:CunninghamExtension.djvu/13

apparatus, the change in the space coordinates will be obscured by the change in the measure of time which will be automatically made.

It may be worth while to give the following forms of the transformation:

$E_{X}=k^{-4}\left(r^{2}-c^{2}t^{2}\right)\left[\left(r^{2}-c^{2}t^{2}\right)e_{x}-2\left\{\left(y^{2}+z^{2}\right)e_{x}+xye_{y}-xze_{z}\right\}-2ct\left(yh_{z}-zh_{y}\right)\right],\dots$ $H_{X}=k^{-4}\left(r^{2}-c^{2}t^{2}\right)\left[-\left(r^{2}-c^{2}t^{2}\right)h_{x}+2\left\{\left(y^{2}+z^{2}\right)h_{x}-xyh_{y}-xzh_{z}\right\}-2ct\left(ye_{z}-ze_{y}\right)\right],$

or, in vector notation,

$E=k^{-4}\left(r^{2}-c^{2}t^{2}\right)\left\{\left(r^{2}-c^{2}t^{2}\right)e+2[r[re]]-2ct[rh]\right\},$ $H=k^{-4}\left(r^{2}-c^{2}t^{2}\right)\left\{-\left(r^{2}-c^{2}t^{2}\right)h-2[r[rh]]-2ct[re]\right\}.$[1]

Invariants and Constitutive Equations under the Transformation.

5. The theorem of relativity being established, it is possible to deduce from any known set of phenomena another which geometrically is the transformation of it. The values of physical quantities in the new phenomena will be obtained from those in the old by the equations obtained in the last section. The results so obtained are quite analogous to those in § 2, and will be given below. Two quantities that are invariant will be mentioned first.^[2]

The following equation is immediately established:

$\left[E_{R}^{2}+E_{\theta }^{2}+E_{\phi }^{2}-H_{R}^{2}-H_{\theta }^{2}-H_{\phi }^{2}\right]=\lambda ^{4}\left[e_{r}^{2}+e_{\theta }^{2}+e_{\phi }^{2}-h_{r}^{2}-h_{\theta }^{2}-h_{\phi }^{2}\right].$

Thus, if l, L denote the differences of the magnetic energies in the two systems per unit volume,

$L=\lambda ^{4}l.$

Now

${\frac {\partial (R,T)}{\partial (r,t)}}={\frac {1}{\lambda ^{2}}}$ and ${\frac {R}{r}}={\frac {1}{\lambda ^{2}}}.$

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1. Mr. Bateman has suggested to me that the formulae of transformation for the scalar and vector potentials are:—
   ${\begin{array}{rl}\Phi &=\beta \lambda \left\{va_{r}/c-\phi \right\},\\A_{R}&=\beta \lambda \left\{a_{r}-v\phi /c\right\},\\A_{N}&=-\lambda a_{n}.\end{array}}$
2. These invariant equations hold equally in the case of the Lorentz-Einstein transformation. Cf. Planck, loc. cit., in reference to the invariance of the kinetic potential.