Page:Electromagnetic phenomena.djvu/5

 As to the coefficient l, it is to be considered as a function of w, whose value is 1 for w=0, and which, for small values of w, differs from unity no more than by an amount of the second order.

The variable t'  may be called the local time; indeed, for k=1, l=1 it becomes identical with what I have formerly understood by this name.

If, finally, we put

these latter quantities being considered as the components of a new vector $$\mathfrak{u}'$$, the equations take the following form:

{{MathForm2|(9)|$$\left.\begin{align} & div'\ \mathfrak{d}'=\left(1-\frac{wu_{x}'}{c^{2}}\right)\varrho',\quad div'\ \mathfrak{h}'=0,\\ & rot'\ \mathfrak{h'}=\frac{1}{c}\left(\frac{\partial\mathfrak{d}'}{\partial t'}+\varrho'\mathfrak{u}\right),\\ & rot'\ \mathfrak{d}'=-\frac{1}{c}\frac{\partial\mathfrak{h}'}{\partial t'}, \end{align}\right\}$$}}

{{MathForm2|(10)|$$\left.\begin{align} & \mathfrak{f}_{x}=l^{2}\mathfrak{d}_{x}^{'}+l^{2}\frac{1}{c}\left(\mathfrak{u}_{y}^{'}\mathfrak{h}_{z}^{'}-\mathfrak{u}_{z}^{'}\mathfrak{h}_{y}^{'}\right)+l^{2}\frac{w}{c^{2}}\left(\mathfrak{u}_{y}^{'}\mathfrak{d}_{y}^{'}-\mathfrak{u}_{z}^{'}\mathfrak{d}_{z}^{'}\right),\\ & \mathfrak{f}_{y}=\frac{l}{k}^{2}\mathfrak{d}_{y}^{'}+\frac{l}{k}^{2}\frac{1}{c}\left(\mathfrak{u}_{z}^{'}\mathfrak{h}_{x}^{'}-\mathfrak{u}_{x}^{'}\mathfrak{h}_{z}^{'}\right)-\frac{l}{k}^{2}\frac{w}{c^{2}}\mathfrak{u}_{x}^{'}\mathfrak{d}_{y}^{'},\\ & \mathfrak{f}_{z}=\frac{l}{k}^{2}\mathfrak{d}_{z}^{'}+\frac{l}{k}^{2}\frac{1}{c}\left(\mathfrak{u}_{x}^{'}\mathfrak{h}_{y}^{'}-\mathfrak{u}_{y}^{'}\mathfrak{h}_{x}^{'}\right)-\frac{l}{k}^{2}\frac{w}{c^{2}}\mathfrak{u}_{x}^{'}\mathfrak{d}_{z}^{'}. \end{align}\right\}$$}}

The meaning of the symbols div'  and rot'  in (9) is similar to that of div and rot in (2); only, the differentiations with respect to x, y, z are to be replaced by the corresponding ones with respect to x', y', z' .

§ 5. The equations (9) lead to the conclusion that the vectors $$\mathfrak{d}'$$ and $$\mathfrak{h}'$$ may be represented by means of a scalar potential φ and a vector potential $$\mathfrak{a}'$$. These potentials satisfy the equations

and in terms of them $$\mathfrak{d}'$$ and $$\mathfrak{h}'$$ are given by