Secondly consider the Lorentzian [or 'electromagnetic'] theory of relativity. The two time-systems for reference to and for reference to respectively are not identical. Let be the measure of the lapse of time in the -system, and be the measure of the lapse of time in the -system. The distinction between the two time-systems is embodied in the fact that event-particles which happen simultaneously at time in -space do not happen simultaneously throughout space . Thus supposing that an event-particle happens at () in -space and -time and at () in -space and -time, we seek for the formulae which are to replace equations (1) of the Newtonian theory. As before let lie in the direction of the motion of in , and in the reverse direction of the motion of in . Also let lie on , so that event-particles which happen on also happen, on . One connection between the two time-systems is secured by the rule that event-particles which happen simultaneously at points in -space on a plane perpendicular to also happen simultaneously at points in -space on a plane perpendicular to . Accordingly the quasi-parallelism of to , and of to , is defined and secured in the same way as for Newtonian relativity. The same meaning as above will be given to and ; also is the fundamental velocity which is the velocity of light in vacuo. Then we define The formulae for transformation are These formulae are symmetrical as between and , so that It is evident that when is small, and when and are not too large Thus the formulae reduce to the Newtonian type. Let , , stand for , etc., and , , , for , etc. Then it follows immediately from the preceding formulae that With the notation of Appendix II to Chapter II, the formulae of transformation for Maxwell's equations are where () is the velocity of the charge at () at the time . Also it immediately follows from formulae (5) that Hence vanish together. This proves Einstein's theorem on the invariance of the velocity , so far as concerns the sufficiency of the Lorentzian formulae to produce that result. CHAPTER IV CONGRUENCE 11. Simultaneity. 11.1 Einstein analysed the ideas of time-order and of simultaneity. Primarily (according to his analysis) time-order only refers to the succession of events at a given place. Accordingly each given place has its own time-order. But these time-orders are not independent in the system of nature, and their correlation is known to us by means of physical measurements. Now ultimately all physical measurement depends upon coincidence in time and place. If and be two places, the time-orders and which belong to and are correlated by observations of coincidences at and at respectively.