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Now there is, as can be easily seen, a certain hyperbola, which has three infinitely contiguous points in common with the world-line at P, and of which the asymptotes are the generators of a fore-cone and an aft-cone. This hyperbola may be called the hyperbola of curvature at P (vide fig. 3). If M be the center of this hyperbola, then we have to deal here with an inter-hyperbola with center M. Let $\varrho$ be the sum of the vector MP, then we perceive that the acceleration-vector at P is a vector of magnitude $c^{2}/\varrho$ in the direction of MP.

If ${\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}$ are nil, then the hyperbola of curvature at P reduces to the straight line touching the world-line at P, and $\varrho =\infty$.

IV.

In order to demonstrate that the assumption of the group $G_{c}$ for the physical laws does not possibly lead to any contradiction, it is inevitable to undertake a revision of the whole of physics on the basis of the assumptions of this group. The revision has, to a certain extent, already been successfully made in the case of thermodynamics and tadiation,"^[1] for electromagnetic phenomena",^[2] and finally for Mechanics with the maintenance of the idea of mass.

For the latter area, the question may be asked: if there is a force with the components X, Y, Z (with respect to the space-axes) at a world-point P(x, y, z, t), where the motion-vector is $\left({\dot {x}},\ {\dot {y}},\ {\dot {z}},\ {\dot {t}}\right)$, then how are we to regard this force when the system of reference is changed in any possible manner? Now, certain well-tested theorems about the ponderomotive force in electromagnetic fields exist, where the group $G_{c}$ is undoubtedly permissible. These theorems lead us to the following simple rule; if the system of reference be changed, then the supposed force is to be put as a force in the new space-coordinates in such a manner, that the corresponding vector with the components

${\dot {t}}X,\ {\dot {t}}Y,\ {\dot {t}}Z,\ {\dot {t}}T,$

where

$T={\frac {1}{c^{2}}}\left({\frac {\dot {x}}{\dot {t}}}X+{\frac {\dot {y}}{\dot {t}}}Y+{\frac {\dot {z}}{\dot {t}}}Z\right)$

is the work of the force divided by $c^{2}$ at the world-point, remains unaltered. This vector is always normal to the motion-vector at P. Such a force-vector belonging to a force at P, may be called a moving force-vector at P.

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1. M. Planck, Zur Dynamik bewegter Systeme, Berliner Ber. 1907, p. 542 (also Ann. d. Phys. 26, 1908, p. 1).
2. H. Minkowski, Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern, Göttinger Nachr. 1908, p. 53.