57.2 Consider an event stationary in the time-system , and let be another time-system. Let be the measure of the normal cross-sections of and let be the measure of the oblique cross-sections made by moments of . We require the ratio of to . Take (as usual) mutual axes for and , and let the event-particle which is the origin lie in which is the antecedent terminal moment of . Then is at the -time zero, and let (the subsequent terminal moment) be at the -time . Then if () be the -coordinates of the event-particle in which a station (of the set composing the event ) intersects , the -coordinates of the other end (the subsequent end) of in are (). Furthermore let () be the -coordinates of the antecedent end of , and let () be the -coordinates of the subsequent end of . Then by the usual formulae [cf. subarticle 52.2] But, by analogous reasoning to that for the elementary case of geometrical parallelograms, the absolute extent of the event can be expressed as and as ). Hence . 57.3 The stations of duration of time-system are portions of points of the time-less space of [the -space]. Thus by prolonging the stations which constitute the stationary event we obtain the assemblage of -points which is the complete assemblage of -points intersecting the cross-sections of , each event-particle in each cross-section lying on one and only one such -point and each of these -points intersecting each cross-section in one event-particle. The assemblage of these -points is a volume of the -space, and the successive instantaneous volumes which are the normal cross-sections of [stationary in ] each occupy this same volume in the -space. Thus the stationary event during the lapse of -time throughout which it endures is happening at the same place in the -space. But the successive oblique cross-sections of formed by moments of another time-system are instantaneous volumes which successively occupy different volumes in the -space. These instantaneous volumes travel in the -space, sweeping over it with the uniform velocity , namely the velocity due to the time-system in the -space. 57.4 A 'normal slice' of a stationary event is the slice of it cut off between any two normal cross-sections. An 'oblique slice' of a stationary event is the slice of it cut off between any two parallel oblique cross-sections. A normal slice of a stationary event is itself a stationary event in the same time-system. 58. Motion of Objects. 58.1 A material object is 'motionless' within a duration when throughout that duration the material object and its extensive components are all situated in stationary events. In the case of a motionless material object, Law I for uniform objects can be made more precise, as follows: If be a material object motionless in the duration and be the stationary event extending throughout in which it is situated, then is situated in any oblique slice of .