Page:AbrahamMinkowski2.djvu/11

 Taking into account the symmetry conditions:

$Y_{z}=Z_{y},\ Z_{x}=X_{z},\ X_{y}=Y_{x},$

we have

$\begin{array}{c} Y'_{z}-Z'_{y}=\mathfrak{q}_{z}\mathfrak{f}_{y}-\mathfrak{q}_{y}\mathfrak{f}_{z},\\ Z'_{x}-X'_{z}=\mathfrak{q}_{x}\mathfrak{f}_{z}-\mathfrak{q}_{z}\mathfrak{f}_{x},\\ X'_{y}-Y'_{x}=\mathfrak{q}_{y}\mathfrak{f}_{x}-\mathfrak{q}_{x}\mathfrak{f}_{y},\end{array}$

relationships that are, as I demonstrated in the first paper, satisfied by the expressions given for relative pressures. Only to prove that, we set the function

$2\Phi'=X'_{x}x^{2}+Y'_{y}y^{2}+Z'_{z}z^{2}+\left(Y'_{z}+Z'_{y}\right)yz+\left(Z'_{x}+X'_{z}\right)zx+\left(X'_{y}+Y'_{x}\right)xy$

equal to

$\begin{array}{ll} 2\Phi' & =2\Phi+x^{2}\mathfrak{q}_{x}\mathfrak{f}_{x}+y^{2}\mathfrak{q}_{y}\mathfrak{f}_{y}+z^{2}\mathfrak{q}_{z}\mathfrak{f}_{z}\\ & +\left(\mathfrak{q}_{y}\mathfrak{f}_{z}+\mathfrak{q}_{z}\mathfrak{f}_{y}\right)yz+\left(\mathfrak{q}_{z}\mathfrak{f}_{x}+\mathfrak{q}_{x}\mathfrak{f}_{z}\right)zx+\left(\mathfrak{q}_{x}\mathfrak{f}_{y}+\mathfrak{q}_{y}\mathfrak{f}_{x}\right)xy,\end{array}$

namely

and introducing the value (24a) of $$2\Phi$$, which is an expression identical to the one resulting from the fundamental formulas ($$V_{a}$$) of the first paper. These finally give

The identity of the values ​​(27) and (27a) will be demonstrated, by proving that the relationship is satisfied:

{{MathForm1|(28)|$$\left\{ \begin{array}{c} (\mathfrak{rq})(\mathfrak{rf})+(\mathfrak{rq})(\mathfrak{rW})\\ \\=(\mathfrak{r,\ E'-E})(\mathfrak{rD})+(\mathfrak{r,\ H'-H})(\mathfrak{rB})-\frac{1}{2}\mathfrak{r}^{2}\left\{ (\mathfrak{E'-E,D})+(\mathfrak{H'-H,B})\right\} \end{array}\right.$$}}

Taking account of (26c) and (25), we can write

Now, it is identically

$\mathfrak{[q,\ B(rD)-D(rB)]=-\left[q[r[DB]]\right]=-r(q[DB])+(rq)[DB],}$|undefined

and the second part of equation (28a) gives in fact:

$(\mathfrak{rq})(\mathfrak{r[DB]})$

so that the relationship (28) is identically satisfied. So, by formula (27a) which is postulated from our system of electrodynamics, the values ​​of the pressures of follow for the special case of the theory of, which obeys the principle of relativity in agreement with relation (24a).