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 the electromagnetic energy density, which were deduced in my cited work from the general principles of electrodynamics of moving bodies.

As a result, these principles are compatible with the postulate of relativity; the symmetry of the electromagnetic field equations for empty space, which is expressed in the Lorentz transformation, should also be given to the electromagnetic equations for ponderable bodies, by writing them down – either in the form of or in the form equivalent to that of  – without contradicting those principles.

has already given to the equations of motion of a material point, a form which is invariant under the Lorentz transformations. However, he thought it necessary to add an additional force to the ponderomotive electromagnetic force, which is incompatible with my system of electrodynamics. In § 4 I will write the equations of motion so that they satisfy the principle of relativity, without introducing the additional force of. However, it must be admitted that the "rest density" of mass is not constant, yet it increases every time, when an electric current generates heat (in Joule) in matter; this hypothesis was already before stated by and  in relation to the principle of relativity.

But it seems doubtful, if the very concept of space and time developed by is a possible basis of rational mechanics. Indeed, the kinematics of rigid systems, which wanted to adapt to the Lorentz group, offers considerable difficulties as shown by  : the rigid body in the "world" of  cannot be set into rotation.

§ 2. Four dimensional vectors.

A linear transformation of the four coordinates $$x, y, z, z$$, which has the invariant

$x^{2}+y^{2}+z^{2}+u^{2}\,$,

is called "Lorentz transformation" according to. We will confine ourselves to the following group of orthogonal transformations, i.e. rotations of a space of four dimensions.

A system of four variables which transform as the coordinates $$x, y, z, u$$, is called a "four-dimensional vector of first kind" $$\left(V_{I}^{4}\right)$$. When projecting into