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

(27)	$2\Phi '=2\Phi +({\mathfrak {rq}})({\mathfrak {rf}})$

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

(27a)

$2\Phi '=({\mathfrak {rE}}')({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {E'D}})+({\mathfrak {rH}}')({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {H'B}})$

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

(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

(28a)

${\begin{cases}({\mathfrak {rq}})({\mathfrak {r[DB]}})&=\left({\mathfrak {r[qB]}}\right)({\mathfrak {rD}})-\left({\mathfrak {r[qD]}}\right)({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}\left\{\left({\mathfrak {D[qB]}}\right)-{\mathfrak {B[qD]}}\right\}\\&={\mathfrak {r[q,\ B(rD)-D(rB)]+r^{2}(q[DB])}}\end{cases}}$

Now, it is identically

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

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 Maxwell follow for the special case of the theory of Minkowski, which obeys the principle of relativity in agreement with relation (24a).