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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

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

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

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

''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 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 ian mechanics and modern electrodynamics automatically disappear. I still shall consider the position of the ian 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 $$m_{1}$$