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 XXXV. Note on the Derivation from the Principle of Relativity of the Fifth Fundamental Equation of the Maxwell-Lorentz Theory.

By, Ph. D., Instructor in Physical Chemistry at the University of Michigan.

we consider two systems of "space time coordinates” S and S' in relative motion in the X direction with the velocity v, any kinematic phenomenon which occurs may be described in terms of the variables x, y, z and t belonging to the system S or z', y', z' and t' belonging to the system S'. The Einstein theory of relativity has led to the following equations for transforming the description of a kinematic phenomenon from one set of coordinates to the other.

(where $$c$$ is the velocity of light and $$\beta$$ is substituted for the fraction $$\tfrac{v}{c}$$).

The content of these equations may be expressed in words, by saying that an observer in the moving system S' (S having been arbitrarily taken as at rest) uses a metre stick which, although the same length as a stationary metre stick when held perpendicular to the line of relative motion of the two systems, is shortened in the ratio of $$\sqrt{1-\beta^{2}}:1$$ when held parallel to OX, that clocks in the moving system beat elf seconds which are longer than those of stationary clocks in the ratio $$1:\sqrt{1-\beta^{2}}$$, and that a clock in the moving system which is $$x'$$ units to the rear of the one at the centre of coordinates is set ahead by $$x'\tfrac{v}{c^{2}}$$ seconds, although the two clocks appear synchronous to the moving observer. A simple non-analytical derivation of these relations has been given in another place.

Let us now take the Maxwell-Hertz equations for the