530 A Definition of Simultaneity and the ^Ether. 



than this. It is obvious that we are justified in speaking 

 quantitatively of the velocity of the phase point R along L, 

 precisely as if we were speaking of a material point or light 

 wave, while the argument employed to demonstrate the 

 uniqueness of the line of instantaneous co-directionality 

 becomes quite definite when considered in connexion with 

 such apparatus. 



We shall call the system of these lines of instantaneous 

 codirectionality, corresponding to the different positions 

 occupied in S by P, the system (LF') and we shall define 

 the events E and E' as simultaneous if both occur in the 

 transit of L across the same line of (LF'). 



To prove that this definition is independent of F, F', and 

 L, we first infer from the definition of instantaneous co- 

 directionality, that if two lines cross one another, as, for 

 example, L and M, and have at all points of this transit 

 finite normal components of relative velocity, then if they 

 are instantaneously codirectional at any point, the line which 

 is the locus of the events of their transit in iiny third frame, 

 will, at that point, be codirectional with both lines. 

 Considering the locus in a third frame of the transit 

 of L across a member of (LF') the definition of simul- 

 taneity is seen to be independent of F', the third frame being 

 interchangeable with F', where the condition of a finite 

 normal velocity relative to L is observed. It is also obvious 

 that F' is interchangeable with F, M taking the place of L. 



We now join P, P' by any second line X in F, and select 

 F' in such a way that the condition of a finite normal com- 

 ponent of relative velocity is satisfied with regard to the 

 closed line L*+X. There will then, through Q, be a unique 

 closed line M + /U in F', which is instantaneously co-direc- 

 tional at all points with L + X, as the latter crosses it, and 

 it is clear that the same criterion of simultaneity is satisfied 

 bv the passage of L + X across M + fi, as by that of L across 

 M or of X across /x, L and X bein^ thus interchangeable and 

 the definition's independence of the particular line L being 

 established. It is clear, further, that if two events are 

 simultaneous with a third, according to this definition, they 

 are also simultaneous with one another, since the line joinino- 

 their locations in F may be chosen to pass through the 

 third. 



The definition is thus absolute, and can be extended 

 throughout the whole universe, while it does not depend 

 in any way upon the Euclidean chaiacter of space. The 

 rather also is that privileged frame of space for which, where 



