Page:AbrahamMinkowski1.djvu/8

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We interpret the vectors \mathfrak{E}',\ \mathfrak{H}' as the forces acting at moving electric and magnetic unit poles. The vectors \mathfrak{D,\ B} we call, by using the terminology of the "Enzyklopädie der mathematischen Wissenschaften", the "electric and magnetic excitation".

It corresponds to the importance of vector \mathfrak{E}', to make that approach for heat, which is developed for the time- and space-unity of moving matter

(III) Q=\mathfrak{JE}'

At this third main equations, a equation is added as the fourth one, which connects the relative ray with the vectors <math>\mathfrak{E}'\mathfrak{H}':

(IV) \mathfrak{S}'=c[\mathfrak{E}'\mathfrak{H}']

For the case of rest, this vector passes into the Poynting vector.

Eventually we need an approach, which expresses the quantity P' defined in equation (13) and by that the relative stresses, by the vectors \mathfrak{E'H'DB}. We put

(V) P'=\mathfrak{E}'(\mathfrak{D}\nabla)\mathfrak{w}+\mathfrak{H}'(\mathfrak{B}\nabla)\mathfrak{w}-\frac{1}{2}[\mathfrak{E'D+H'B}\}\mathrm{div}\mathfrak{w}

and thus we obtain for the relative stresses:

(Va) \begin{cases}
X'_{x}=\mathfrak{E}'_{x}\mathfrak{D}_{x}+\mathfrak{H}'_{x}\mathfrak{B}_{x}-\frac{1}{2}[\mathfrak{E'D+H'B}\}\\
X'_{y}=\mathfrak{E}'_{x}\mathfrak{D}_{y}+\mathfrak{H}'_{x}\mathfrak{B}_{y},\\
X'_{z}=\mathfrak{E}'_{x}\mathfrak{D}_{z}+\mathfrak{H}'_{x}\mathfrak{B}_{z};\\
Y'_{x}=\mathfrak{E}'_{y}\mathfrak{D}_{x}+\mathfrak{H}'_{y}\mathfrak{B}_{x},\\
Y'_{y}=\mathfrak{E}'_{y}\mathfrak{D}_{y}+\mathfrak{H}'_{y}\mathfrak{B}_{y}-\frac{1}{2}[\mathfrak{E'D+H'B}\},\\
Y'_{z}=\mathfrak{E}'_{y}\mathfrak{D}_{z}+\mathfrak{H}'_{y}\mathfrak{B}_{z};\\
Z'_{x}=\mathfrak{E}'_{z}\mathfrak{D}_{x}+\mathfrak{H}'_{z}\mathfrak{B}_{x},\\
Z'_{y}=\mathfrak{E}'_{z}\mathfrak{D}_{y}+\mathfrak{H}'_{z}\mathfrak{B}_{y},\\
Z'_{z}=\mathfrak{E}'_{z}\mathfrak{D}_{z}+\mathfrak{H}'_{z}\mathfrak{B}_{z}-\frac{1}{2}[\mathfrak{E'D+H'B}\}.\end{cases}

For the case of rest, the known formulas for the fictitious stresses follow from that.

The choice of expressions (IV) and (V) appears to be at first sight as totally arbitrary. Yet it is the simplest generalization of the laws valid in resting bodies, which only uses the vectors occurring in the two first main-equations.

Incidentally, it follows from (Va):

Y'_{x}-X'_{y}=\mathfrak{D}_{x}\mathfrak{E}'_{y}-\mathfrak{D}_{y}\mathfrak{E}'_{x}+\mathfrak{B}_{x}\mathfrak{H}'_{y}-\mathfrak{B}_{y}\mathfrak{H}'_{x}.

According to this, the torque of relative stresses is:

(Vb) \mathfrak{R}'=[\mathfrak{DE}']+[\mathfrak{BH}']

The mechanical principles laid out in the previous paragraph, and the fife main-equations, are the foundations upon which our system of electrodynamics of moving bodies is resting.