Page:An Enquiry Concerning the Principles of Natural Knowledge.djvu/173

 MOTION 159 -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 e will be called the * absolute extent of e. 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 K be any function of (x ya y a, z ya t a ), it can by (ii) of 52-2 be also expressed as a function of (x^yo, *p ,pt), and then by (i) Isuzu a yd a dz a st a = IMKdxpdypdZfidtp. . . .(ii), 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 K is an invariant form for the equations of trans formation 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 formula of this sub article hold of each type of kinematics. -4 The hyperbolic type of kinematics has issued in the formula of the Armor-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. The physical meaning of c is also well known; namely, any velocity which in any time-system is of