Page:AbrahamMinkowski2.djvu/10

Instead of (21a, b, c), thus the formulas follow:

(24a)

${\begin{cases}2\Phi &=X_{x}x^{2}+Y_{y}y^{2}+Z_{z}z^{2}+2Y_{z}yz+2Z_{x}zx+2X_{y}xy\\&=({\mathfrak {rE}})({\mathfrak {rD}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {ED}})+({\mathfrak {rH}})({\mathfrak {rB}})-{\frac {1}{2}}{\mathfrak {r}}^{2}({\mathfrak {HB}})+({\mathfrak {rq}})({\mathfrak {rW}}),\end{cases}}$

(24b)

$2{\mathfrak {f}}=[{\mathfrak {EH}}]+[{\mathfrak {DB}}]-{\mathfrak {W}}-{\mathfrak {q}}({\mathfrak {qW}})$

(24c)

$2\psi ={\mathfrak {ED}}+{\mathfrak {HB}}-2({\mathfrak {qW}}).$

These values ​​are identical to those values (in the case of Minkowki's theory) derived from my system of electrodynamics of moving bodies in the first paper.

In this theory, we apply the relationship:^[1]

(25)	${\begin{cases}{\mathfrak {D}}=\epsilon {\mathfrak {E}}'-[{\mathfrak {qH}}],&{\mathfrak {E}}={\mathfrak {E}}'-[{\mathfrak {qB}}],\\{\mathfrak {B}}=\mu {\mathfrak {H}}'+[{\mathfrak {qE}}],&{\mathfrak {H}}={\mathfrak {H}}'+[{\mathfrak {qD}}].\end{cases}}$

A calculation, not reproduced here, gives us

$[{\mathfrak {DB}}]-[{\mathfrak {EH}}]=k^{-2}(\epsilon \mu -1)[{\mathfrak {E}}'{\mathfrak {H}}']=k^{-2}(\epsilon \mu -1){\mathfrak {f}}'.$

On the other hand we have, according to (22):

(26)	${\mathfrak {W}}-{\mathfrak {q}}({\mathfrak {qW}})=k^{-2}(\epsilon \mu -1){\mathfrak {f}}'.$

So the following relation holds:

(26e)

${\mathfrak {W}}-{\mathfrak {q}}({\mathfrak {qW}})=[{\mathfrak {DB}}]-[{\mathfrak {EH}}],$

a formula already present in the first paper.^[2]

Equation (24b) therefore can be written

(26b)

${\mathfrak {f}}=[{\mathfrak {EH}}]-{\mathfrak {q}}({\mathfrak {qW}})$

namely

(26c)

${\mathfrak {f}}=[{\mathfrak {DB}}]-{\mathfrak {W}}.$

Evidently from (26b) and (18b), the energy current postulated by the Poynting theorem follows in the case of rest. The values ​​of the energy current and momentum density are consistent with those found in the first paper^[3]. Even the expression (24c) of the energy density was already indicated there^[4].

It remains to prove that the electromagnetic pressures, determined from equation (24a), are those that result from the first paper.

To prove this, we must introduce the "relative pressure", defined by^[5]

${\begin{array}{ccccc}X'_{x}=X_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{x},&&X'_{y}=X_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{x},&&X'_{z}=X_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{x};\\Y'_{x}=Y_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{y},&&Y'_{y}=Y_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{y},&&Y'_{z}=Y_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{y};\\Z'_{x}=Z_{x}-{\mathfrak {q}}_{x}{\mathfrak {f}}_{z},&&Z'_{y}=Z_{y}-{\mathfrak {q}}_{y}{\mathfrak {f}}_{z},&&Z'_{z}=Z_{z}-{\mathfrak {q}}_{z}{\mathfrak {f}}_{z}.\end{array}}$

1. M. Abraham, l. c., equations (36) and (37).
2. M. Abraham, l. c. equation (40c).
3. M. Abraham, l. c. equations (40), (40a) and (42).
4. M. Abraham, l. c. equation (44a).
5. M. Abraham, l. c. equation (10), where it has to be put ${\mathfrak {w}}=c{\mathfrak {q}},\ {\mathfrak {f}}=c{\mathfrak {g}}$.