528 Mr. E. H. Synge on a Definition of 



the existing apparatus are explained. The agreement between 

 theory and experiment is shown to be good. 



The questions o£ improvement in position o£ the plate and 

 second-order focussing are discussed. The latter is shown 

 to be feasible. 



Lines on which future improvements are possible are 

 suggested. 



Trinity College, Cambridge. 

 November 1921. 



LX. ..4 Definition of Simultaneity and the ^Ether. 

 By E. H. Synge*. 



I PROPOSE to show that a definition of absolute simul- 

 taneity is involved in the generalized coordinate frames 

 assumed in relativity. 



Considering two events E, W , located at the points P, P', 

 in an arbitrary frame F, and joining P, P' by any line L in F, 

 we may assign, in an infinite number of ways, another frame 

 F', which, at every instant, at every point of L, has a finite 

 component of relative velocity .normal to L. The points of 

 L trace out a surface S in F', and if Q be the point in S 

 occupied by P at a specific instant, and if M be any line in 

 S passing through Q, then the point R in L, at which M and 

 L instantaneously intersect as L crosses M, will move along 

 L as L moves in S, and in the immediate neighbourhood of 

 P this motion of R along I/will, according to the direction 

 of M in S at Q, be either inwards to, or outwards from, P. 

 The possible directions of M at Q are divisible into two 

 groups, corresponding to these two different senses of the 

 motion of R, and it is clear^ on grounds of continuity, that 

 the two groups in question are separated by a limiting 

 direction of discontinuity, for which the motion of P along 

 L at P, and obviously also of R along M at Q, would be 

 indeterminate in sense, and of infinite velocity. 



We shall define this limiting direction as instantaneously 

 codirectional with L at P, and, considering the entire length 

 of L, it appears without difficulty, in the supposed circum- 

 stances of motion, that through each position instantaneously 

 occupied by P in S, a single line in S can be drawn, which 

 at all points of its length is instantaneously codirectional with 

 L as the latter crosses it. For, supposing, on the contrary, 

 that two such lines passed through a point Q' in S, and 



* Communicated by the Author. 



