The first part of the principle can be enunciated as the statement that the measures of relative velocities [i.e. the velocity of in and of in ] are equal and opposite; namely The second part is the principle of the symmetry of two time-systems in respect to transverse velocities; namely, if a velocity in , normally transverse to the direction of in , is represented by the velocity () in , where is along the direction of in and ′ is normally transverse to it, then the same magnitude of velocity in , normally transverse to the direction of in , is represented by the velocity () in , where is along the direction of in , and ′ is normally transverse to it. From the first part of the principle, by (ii) and (iii) of 50.1, we deduce In order to apply the second part of the principle we first identify with ), then from (i) and (ii) of 50.1 Again we identify ( with ), and by interchanging and in the above formulae we find Hence by the second part of the principle 51. Transitivity of Congruence. 51.1 It follows, from (iii) of 49.7, and from (ii) and (iii) of 50.1, and from (i), (ii), (iii) of 50.3, that equations (i) of 50.1 can be written We can now express and in terms of and an absolute constant by considering deductions from the transitivity of congruence. 51.2 Let be a time-system such that the level contains and , and let these rects be the axes and . Then the matrix contains , , and . Thus we have obtained a set of mutual axes for and ; namely, () and (), where and now play the part that and sustain for and . Thus the velocities of the time-system in and are, by (i) of 51.1, connected by We have here assumed the congruence of the time-units in and . Now identify and . Then Hence from (i) of 51.1 Again identify with . Then Hence from (i) of this subarticle From (ii) and (iii) and (i) of 50.3 51.3 Evidently if be any other member of the collinear set of time-systems (, ), then Hence if be a collinear set of time-systems, and , , , be any four of its members, and hence, since , we obtain where is a constant for the collinear set. Furthermore, if be a time-system not belonging to but related to and as explained in 51.2, 51.4 Now let , , be any three non-collinear time-systems, and construct a diagram to represent elements in the time-less space of according to the familiar method of geometricians. The points of the diagram symbolise -points, and the straight lines of the diagram symbolise -lines. Let be any -point and let be the direction in -space of the velocity . Then is the direction in -space of the velocity (positive or negative) of any member of the collinear set (). Fig. 18.