this follows from the second of equations (29) and shows that the clock goes slower than if it were at rest relatively to '. These two consequences, which hold, mutatis mutandis, for every system of reference, form the physical content, free from convention, of the Lorentz transformation. Addition Theorem for Velocities. If we combine two special Lorentz transformations with the relative velocities and , then the velocity of the single Lorentz transformation which takes the place of the two separate ones is, according to (27), given by [Pg 38] General Statements about the Lorentz Transformation and its Theory of Invariants. The whole theory of invariants of the special theory of relativity depends upon the invariant (23). Formally, it has the same rôle in the four-dimensional space-time continuum as the invariant + + in the Euclidean geometry and in the pre-relativity physics. The latter quantity is not an invariant with respect to all the Lorentz transformations; the quantity of equation (23) assumes the rôle of this invariant. With respect to an arbitrary inertial system, may be determined by measurements; with a given unit of measure it is a completely determinate quantity, associated with an arbitrary pair of events. The invariant differs, disregarding the number of dimensions, from the corresponding invariant of the Euclidean geometry in the following points. In the Euclidean geometry is necessarily positive; it vanishes only when the two points concerned come together. On the other hand, from the vanishing of it cannot be concluded that the two space-time points fall together; the vanishing of this quantity , is the invariant condition that the two space-time points can be connected by a light signal in vacuo. If a point (event) represented in the four-dimensional space of the , , , then all the "points" which can be connected to by means of a light signal lie upon the cone = 0 (compare Fig. 1, in which the dimension is suppressed). The "upper" half of the cone may contain the "points" to which light signals can be sent from ; then the "lower" half [Pg 39] of the cone will contain the "points" from which light signals can be sent to . The points enclosed by the conical surface furnish, with , a negative ; as well as is then, according to Minkowski, of the nature of a time. Such intervals represent elements of possible paths of motion, the velocity being less than that of light.[9] In this case the -axis may be drawn [Pg 40] in the direction of by suitably choosing the state of motion of the inertial system. If lies outside of the "light-cone" then is of the nature of a space; in this case, by properly choosing the inertial system, can be made to vanish. FIG. 1. [9]That material velocities exceeding that of light are not possible, follows from the appearance of the radical in the special Lorentz transformation (29).