Manchester Memoirs, Vol. liv. (1910), No. 14. 5 



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^,...A^, 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* 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.f 



* An analytical proof of this result is given l)y Prof A. W. Conway. 

 jR/Ci-. London Math. Sac, Ser. 2, vol. i. (1903). 



t If the times associited with B and A in two views of them are /| and 

 t„ respectively, B will be able to witness at time /j an event experienced by 

 B at time t„ if a sphere of radius ct-^ having the point B as centre is touched 

 internally by a sphere of radius ct^ having the point A as centre. 



Now if B is moving with a velocity less than that of light, the spheres 

 associated w ilh consecutive positions of B surround one another in succession 

 as in Fig. i. 



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



Fiz. I. 



