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that the world-line of a second point-electron passes through the world-point $P_{1}$. Let us determine P, Q, r as before, construct the middle-point of the hyperbola of curvature at P, and finally the normal MN from M upon a line through P which is parallel to $QP_{1}$. With P as the initial point, we shall establish a system of reference in the following way: the t-axis will be laid along PQ, the x-axis in the direction of $QP_{1}$; the y-axis in the direction of MN, then the direction of the z-axis is automatically determined as normal to the t-,x-,y--axes. Let ${\ddot {x}},\ {\ddot {y}},\ {\ddot {z}},\ {\ddot {t}}$ be the acceleration-vector at $P$, ${\dot {x}}_{1},\ {\dot {y}}_{1},\ {\dot {z}}_{1},\ {\dot {t}}_{1}$ be the motion-vector at $P_{1}$. Then the force-vector exerted by the first electron e (moving in any possible manner) upon the second electron $e_{1}$ (likewise moving in any possible manner) at $P_{1}$ is represented by

$-ee_{1}\left({\dot {t}}_{1}-{\frac {{\dot {x}}_{1}}{c}}\right){\mathfrak {K}}$

For the components ${\mathfrak {K}}_{x},\ {\mathfrak {K}}_{y},\ {\mathfrak {K}}_{z},\ {\mathfrak {K}}_{t}$ of the vector ${\mathfrak {K}}$ the three relations hold: —

$c{\mathfrak {K}}_{t}-{\mathfrak {K}}_{x}={\frac {1}{r^{2}}},\ {\mathfrak {K}}_{y}={\frac {\ddot {y}}{c^{2}r}},\ {\mathfrak {K}}_{z}=0$

and fourthly this vector ${\mathfrak {K}}$ is normal to the motion-vector $P_{1}$, and through this circumstance alone, its dependence on this latter motion-vector arises.

If we compare with this expression the previous formulations^[1] giving the same elementary law about the ponderomotive action of moving electric point-charges upon each other, then we cannot but admit, that the relations which occur here only reveal the inner essence of full simplicity first in four dimensions; but upon a space of three dimensions that is forced upon them from the outset, they cast very complicated projections.

In the mechanics reformed according to the world-postulate, the disharmonies which have disturbed the relations between Newtonian mechanics and modern electrodynamics automatically disappear. I still shall consider the position of the Newtonian law of attraction to this postulate. I will assume that when two point-masses m and $m_{1}$ describe their world-lines, a moving force-vector is exerted by m upon $m_{1}$, and the expression is just the same as in the case of the electron previously discussed; we only have to write $+mm_{1}$ instead of $-ee_{1}$. We shall consider now the special case in which the acceleration-vector of m is constantly zero; then t may be introduced in such a manner that m may be regarded as fixed, the motion of $m_{1}$
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1. K. Schwarzschild, Göttinger Nachr. 1903, p. 132. — H. A. Lorentz, Enzykl. d. math. Wissensch., Art. V, 14, p. 199.