I shall now describe the ponderomotive force which is exerted by one moving electron upon another moving electron. Let us suppose that the world-line of a second point-electron passes through the world-point P₁. Let us determine P, Q, r as before, construct the middle-point M of the hyperbola of curvature at P, and finally the normal MN upon a line through P which is parallel to QP₁. 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 QP₁. The y-axis in the direction of MN, then the z-axis is automatically determined, as it is normal to the x-, y-, z-axes. Let [:x], [:y], [:z], [:t] be the acceleration-vector at P, [.x]₁, [.y]₁ [.z]₁, [.t]₁ be the velocity-vector at P₁. Then the force-vector exerted by the first election e, (moving in any possible manner) upon the second election e, (likewise moving in any possible manner) at P₁ is represented by
-e e₁([.t₁] - [.x₁]/c)F,
For the components Fx, Fy, Fz, Ft of the vector F the following three relations hold:—
cFt - Fx = 1/r², Fy = [:y]/(c²r), Fz = 0,
and fourthly this vector F is normal to the velocity-vector P₁, and through this circumstance alone, its dependence on this last velocity-vector arises.
If we compare with this expression the previous formulæ[[35]] giving the elementary law about the ponderomotive action of moving electric charges upon each other, then we cannot but admit, that the relations which occur here reveal the inner essence of full simplicity first in four dimensions; but in three dimensions, they have 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 shall now consider the position of the Newtonian law of attraction to this postulate. I will assume that two point-masses m and m₁ describe their world-lines; a moving force-vector is exercised by m upon m₁, and the expression is just the same as in the case of the electron, only we have to write +mm₁ instead -ee₁. We shall consider only the special case in which the acceleration-vector of m is always zero: then t may be introduced in such a manner that m may be regarded as fixed, the motion of m is now subjected to the moving-force vector of m alone. If we now modify this given vector by writing -([.]1/√(1-(v²/c²)) instead of [.t] ([.t] = 1 up to magnitudes of the order (1[.]/c²)), then it appears that Kepler’s laws hold good for the position (x₁, y₁, z₁), of m₁ at any time, only in place of the time t₁, we have to write the proper time τ₁ of m₁. On the basis of this simple remark, it can be seen that the proposed law of attraction in combination with new mechanics is not less suited for the explanation of astronomical phenomena than the Newtonian law of attraction in combination with Newtonian mechanics.
Also the fundamental equations for electro-magnetic processes in moving bodies are in accordance with the world-postulate. I shall also show on a later occasion that the deduction of these equations, as taught by Lorentz, are by no means to be given up.
The fact that the world-postulate holds without exception is, I believe, the true essence of an electromagnetic picture of the world; the idea first occurred to Lorentz, its essence was first picked out by Einstein, and is now gradually fully manifest. In course of time, the mathematical consequences will be gradually deduced, and enough suggestions will be forthcoming for the experimental verification of the postulate; in this way even those, who find it uncongenial, or even painful to give up the old, time-honoured concepts, will be reconciled to the new ideas of time and space,—in the prospect that they will lead to pre-established harmony between pure mathematics and physics.