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occurrence, if space is of three dimensions. If there are more than four observers in the standard system the relations between all the different observations of an event will depend upon the nature of space, and will take a comparatively simple form if the space is Euclidean.

If the position and time associated with an object B is always determined from measurements by a number of standard observers $A_{1},\dots A_{r}$, so that a consistent universal time exists for each point of space, the following conclusion may be deduced by elementary geometry for Euclidean space:

If two observers B and C are at rest or in motion relative to the standard system, and their velocities are less than that of light, there is only one instant^[1] at which B is able to observe an instantaneous event experienced by C, but if one of the observers is moving with a velocity greater than that of light this is not necessarily the case; in fact it may happen that B sees two or more pictures of the same event.^[2]

1. An analytical proof of this result is given by Prof. A. W. Conway. Proc. London Math. Soc., Ser. 2, vol. 1. (1903).
2. If the times associated with B and A in two views of them are $t_{1}$ and $t_{2}$ respectively, B will be able to witness at time $t_{1}$ an event experienced by B at time $t_{2}$ if a sphere of radius $ct_{1}$ having the point B as centre is touched internally by a sphere of radius $ct_{2}$ having the point A as centre.

   Now if B is moving with a velocity less than that of light, the spheres associated with consecutive positions of B surround one another in succession as in Fig. 1.

   It is clear then that there is only one sphere of the series which is touched