Page:AbrahamMinkowski2.djvu/7

Then they become:

(16)

${\begin{cases}{\mathfrak {K}}_{x}={\frac {\partial X_{x}}{\partial x}}+{\frac {\partial X_{y}}{\partial y}}+{\frac {\partial X_{z}}{\partial z}}+{\frac {\partial X_{u}}{\partial z}},\\\\{\mathfrak {K}}_{y}={\frac {\partial Y_{x}}{\partial x}}+{\frac {\partial Y_{y}}{\partial y}}+{\frac {\partial Y_{z}}{\partial z}}+{\frac {\partial Y_{u}}{\partial u}},\\\\{\mathfrak {K}}_{z}={\frac {\partial Z_{x}}{\partial x}}+{\frac {\partial Z_{y}}{\partial y}}+{\frac {\partial Z_{z}}{\partial z}}+{\frac {\partial Z_{u}}{\partial u}},\\\\{\mathfrak {K}}_{u}={\frac {\partial U_{x}}{\partial x}}+{\frac {\partial U_{y}}{\partial y}}+{\frac {\partial U_{z}}{\partial z}}+{\frac {\partial U_{u}}{\partial u}}.\end{cases}}$

Now, in Minkowski's theory, the system of four variables

${\mathfrak {K}}_{x},\ {\mathfrak {K}}_{y},\ {\mathfrak {K}}_{z},\ {\mathfrak {K}}_{u}$,

where the first three are the components of $V^{3}$ which determines the ponderomotive force per unity of space, must form a $V_{I}^{4}$. The 16 variables $X_{x},\ X_{y},\dots ,U_{z},\ U_{u}$, from which that system is derived using equations (16), must be transformed in such a way that this condition is satisfied.

We determine these 16 variables as follows. They are reduced to 10 ones:

(17)	$\left\{{\begin{array}{ccccccc}X_{x},\ Y_{y},\ Z_{z},&&Y_{z}=Z_{y},&&Z_{x}=X_{z},&&X_{y}=Y_{x};\\X_{u}=U_{x},&&Y_{u}=U_{y},&&Z_{u}=U_{z},&&U_{u},\end{array}}\right.$

which are the components of a $T^{4}$.

Then, from the transformation properties of the components of a $T^{4}$ and the components of operator 'lor', it follows that the four variables, derived from (16), transform as the coordinates $x,y,z,u$ of a point of a four-dimensional space, i.e. as components of a $V_{I}^{4}$, in agreement with the principle of relativity. But this determination chosen by me is not the only one which corresponds to this principle. Indeed, Minkowski himself preferred a different determination, which does not satisfy the symmetry conditions contained in (17). Yet the determination postulated from my system of electrodynamics of moving bodies is demonstrated now.

Our aim is to form a $T^{4}$, whose components correspond to the expressions given in the first paper for the special case of Minkowski's theory. If this is established, it is clear that these expressions satisfy the principle of relativity.

We obtain such a $T^{4}$ by calculating a $S^{4}$ in the form of (13), i.e. a homogeneous quadratic function of $x,y,z,u$, which is invariant under a Lorentz transformations:

(18)	$\varphi (x,y,z,u)=\Phi (x,y,z)-iu({\mathfrak {rf}})+{\frac {1}{2}}u^{2}\psi$

From $S^{3}$, which is a second-order homogeneous function of $x,y,z$:

(18a)

$\Phi (x,y,z)={\frac {1}{2}}X_{x}x^{2}+{\frac {1}{2}}Y_{y}y^{2}+{\frac {1}{2}}Z_{z}z^{2}+Y_{z}yz+Z_{x}zx+X_{y}xy$

we obtain the six pressures of Maxwell; a $V^{3}$ which is now designated by ${\mathfrak {f}}$, gives us at the same time (according to (15ab)) the energy current ${\mathfrak {S}}$ and the electromagnetic momentum density