50. The Principle of Kinematic Symmetry. 50.1 Consider any other time-system . The -point () occupied by () and the -point occupied by the same event-particle lie on a matrix which includes an -line () of which every -point is intersected by . Thus correlates the -point () with the -time and the neighbouring -point on namely (), with the neighbouring -time . In this way makes every set of -coordinates of a variable -point to be a function of ; namely it correlates an -point () with the velocity (), which can also be written Analogously the same time-system correlates a -point () with the velocity () which can be written Now the time-system indicates a definite transference from an event-particle () to another event-particle () occupying the same -point (), where any mutually normal -coordinates are employed. The former event-particle is that indicated by () and by (), and the latter event-particle by () and by (). Hence from equations (i) of 49.7 Now is any time-system. First identify it with . Then . Hence , and is the velocity of the time-system in the space of [or, more briefly, the 'velocity of in ']. Let this velocity be ; it is evidently along the -axis in the space of , and Again identify the system with . Then ; and hence , and is the velocity of in . Let this velocity be ; it is along the -axis in the space of , and 50.2 We will now introduce what we will term the 'Principle of Kinematic Symmetry.' Before enunciating this principle it is necessary to determine a standard method of choosing the positive directions of the axes and in the matrix and of the axes and . By reference to the figure of subarticle 45.2 it will be seen that, of the four angular regions into which the rects and divide the matrix , two vertically opposite regions include no point-tracks passing through and the remaining two such regions include point-tracks as well as rects through . The standard choice of positive directions for and is such that the two regions bounded one by both positive directions of these axes, and the other by both negative directions, should include only rects passing through . The positive directions for and are settled by the rule that a positive measure of lapse of time should indicate subsequence in the time-order to the moment . This rule is definite because of the ultimate distinction between antecedence and subsequence in time, which has not otherwise been made use of. This standard choice of positive directions along mutual axes for two time-systems will always be adopted. 50.3 The principle of kinematic symmetry has two parts, enunciating consequences which flow from the fact that the time-units in two time-systems and are congruent. The first part may be taken as the definition, or necessary and sufficient test, of such congruence.