Page:CarmichealPostulates.djvu/20

 and time coordinates. Let two systems of reference S and $$S'$$ have the relative velocity v in the line l. Let systems of rectangular coordinates be attached to the systems of reference S and $$S'$$ in such way that the x-axis of each system is in the line l, and let the y-axis and the z-axis of one system be parallel to the y-axis and the z-axis respectively of the other system. Let the origins of the two systems coincide at the time t = 0. Furthermore, for the sake of distinction, denote the coordinates on S by x, y, z, t and those on $$S'$$ by x', y', z', t'. We require to find the value of the latter coordinates in terms of the former.

From postulate L it follows at once that y' = y and z' = z. Let an observer on S consider a point which at the time t = 0 appears to him to be at distance x from the $$y'z'$$-plane; at time t = t it will appear to him to be at the distance x - vt from the $$y'z'$$-plane. Now, by an observer on $$S'$$ this distance is denoted by $$x'$$. Then from theorem, VI. we have

Now consider a point at the distance x from the yz-plane at time t = t in units of system S. From theorem VII. it follows that to an observer on S the clock on $$S'$$ at the same distance x from the yz-plane will appear behind by the amount

where c is the velocity of light. That is, in units of S this clock would register the time

Hence, by means of theorem IV., we have at once the result

Solving the two equations involving $$x'$$ and $$t'$$ and collecting results, we have

where $$\beta=v/c$$ and c is the velocity of light.

In the same way we may obtain the equations which express t, x, y, z