Page:CarmichealPostulates.djvu/13

 parallel; $$S_1$$ and $$S_2$$ have a relative velocity v ; S and $$S_1$$ have a relative velocity ½v in one sense and S and $$S_2$$ have a relative velocity ½v in the opposite sense. The system S consists of a single light-source, and this source is symmetrically placed with respect to two points of which one is fixed to $$S_1$$ and the other is fixed to $$S_2$$. This is possible as a permanent relation on account of the relative motions of the three systems. For convenience, let us assume S to be at rest.

We shall now suppose that observers on the systems $$S_1$$ and $$S_2$$ measure the velocity of light as it emanates from the source S. Let a point A in $$S_1$$ and a point B in $$S_2$$ which are symmetrically placed with respect to the light-source S, move along the lines $$l_1$$ and $$l_2$$; these lines are parallel. From postulate L it follows that observers on $$S_1$$ and $$S_2$$ will obtain the same measurement of the distance between $$l_1$$ and $$l_2$$. Denote this distance by d. On account of postulate M neither observer is able to detect his motion. Therefore he will make his observations on the assumption that his system is at rest; that is to say, his measurements will be made by means of the units belonging to his system and no corrections will be made on account of the motion of the system. Let the observer on $$S_1$$ reflect a beam of light SA from the point A to a point C on $$l_w$$ and back to A; and let the observed time of passage of the light from A to C and back to A be t. Since the observer assumes his system to be at rest he will suppose that the ray of light passes (in both directions) along the line AC which is perpendicular to $$l_1$$ and $$l_2$$. His measurement of the distance traversed by the ray of light in time t will therefore be 2d. Hence he will obtain as a result

where c is his observed velocity of light.

Similarly, an observer on $$S_2$$, supposing his system to be at rest, finds the time $$t_1$$ which it requires for a ray of light to pass from B to D and return, the ray employed being gotten by reflecting a ray SB at B. Thus the second observer obtains the result

where $$c_1$$ is his observed velocity of light.