49.5 Let be any event-particle in the moment , and let occupy the -point which intersects the moment in the event-particle . Let the lapse of time between the moments and be , where is positive when is subsequent to ; and let the coordinates of the -point in the -space be (). Then the coordinates of in the momentary space of and of in the momentary space of are also (). Also the '-coordinates' of in the four-dimensional space of particles are (); this fact for can also be expressed by saying that occupies the -point () at the -time . A moment, viewed as a locus of event-particles, is represented by a linear equation in the four coordinates (). But the converse is not true; namely, not every linear equation represents a moment. A pair of linear equations represent a level or a matrix, and three independent linear equations represent a rect or a point-track or a null-track. 49.6 If and be any two time-systems, two sets of mutually normal axes, , and , can be found as in the previous subarticle. But these two sets can evidently be so adjusted that is identical with and is identical with , where the two rects () must both lie in the level . Fig. 17. Then the matrix normal to this level at will be denoted by ; it contains through one -point , one -point one -rect , and one -rect . Then any event-particle is referred to the axes for the system , and to the axes for the system . Let its -coordinates be () and its -coordinates be (), where , and . In the diagram, for the sake of simplicity, the particle is in the matrix ; and its coordinates [as in the diagram] in the two systems are () and (), where (with its proper sign) is , (with its proper sign) is , (with its proper sign) is , and (with its proper sign) is . A pair of sets of four axes for a and allied as described in this subarticle are called 'mutual axes' for the two systems. 49.7 The formulae for transformation from the a-coordinates to the -coordinates, referred to mutual axes, are obviously of the form where are constants dependent on the two systems and and on the two arbitrarily chosen units of time-lapse in and , but evidently not dependent on the arbitrarily chosen set of rectangular rects and in the level . The corresponding ()-equations, interchanging and , are The two pairs of ()-equations, (i) and (ii), must be equivalent. The conditions are Only four out of these five conditions are independent.