Let be the direction in -space of the velocity ; by hypothesis is distinct from . Let be the -line perpendicular to the -plane , and let be a time-system whose velocity in , namely , is along . Let denote the collinear set (), ′ the collinear set (), and ″ the collinear set (). Hence from (vi) of 51.3 Hence from (vii) of 51.3 Hence, since and , it is easy to prove that is the same for any pair of time-systems; in other words, that is an absolute constant. 52. The Three Types of Kinematics. 52.1 There are thus three types of kinematics possible, according as is positive, negative, or infinite. The formally possible type where is zero requires that either or should be zero; by reference to (i) of 49.7 and to (i) of 51.1 this supposition is seen to lead to results in such obvious contradiction to experience as to preclude the necessity for further examination. Let us name the types retained (according to the familiar habit) the 'hyperbolic,' the 'elliptic' and the 'parabolic' types of kinematics. 52.2 First consider the hyperbolic type and put for . The equations of articles 49 and 51 then become The equations of transformation, namely (ii), can be expressed symmetrically as between and by means of the scheme [where ] , , , , 0, 0, 0, 1, 0, 0 0, 0, 1, 0 0, 0, 52.3 We notice that The integral taken throughout the four-dimensional region of the set of event-particles which analyse [cf. 37.3] an event will be called the 'absolute extent' of . It follows from (i) that the absolute extent of an event is independent of the time-system in which its measure is expressed. Furthermore if be any function of (), it can by (ii) of 52.2 be also expressed as a function of (), and then by (i) or, in more familiar form, where the limits are taken to include some event. We may expect important physical properties to be expressible in terms of such integrals, in particular where is an invariant form for the equations of transformation of 52.2, and when the conditions, which the quantity represented by the integral satisfies, are also invariant in their expression in different time-systems. The formulae of this subarticle hold of each type of kinematics. 52.4 The hyperbolic type of kinematics has issued in the formulae of the Larmor-Lorentz-Einstein theory of electromagnetic relativity, namely, the theory by which with a certain amount of interpretation the electromagnetic equations are invariant for these transformations.