388 Leigh Page, 



involved in the original relativity principle as Euclidean systems in 

 which light travels in straight lines with constant speed c.^ 



The simplest type of reference system is that in which light 

 travels with the same speed c at all points, all times, and in all 

 directions. A more general type is that in which the speed C 

 of, light is the same in all directions at any one point and time 

 but differs from point to point and from time to time. Subse- 

 quent consideration will be limited to systems of this degree of 

 generality, since no experiment has yet indicated the need of 

 referring" observations to a still more generalized type of system. 



Let 6" and S' be two systems of the degree of generality just 

 defined. Unprimed letters will refer to quantities in 5 and 

 primed letters to those in S'. Let O' and P' be two nearby points 

 fixed in S'. Let O be the position of O' and P that of P' in 6^ 

 at the time t. At this time let a light signal be despatched from 

 O toward P'. Let P^ be the position of P' and t -\- dt^ the time 

 in vS" when the signal reaches this point. At the instant of arrival 

 of the signal at P^ let a return signal be despatched toward O'. 

 Let O' be at 0^ in vS" when the return signal reaches it, and denote 

 the time of this event in 6^ by ^ -|- dtj^ -\- dfn. Since the speed of 

 light at any point in S' is not a function of direction, the times 

 dt^' and dto taken by the light signal to pass from O' to P' and 

 back again will be equal. 



Denote by dr^ and d)\' the distances OP and O'P'. With O 

 as origin choose rectangular axes^ XYZ in 6^ so that the X axis 

 is parallel to the velocity v of O' relative to S. Denote the co- 

 ordinates of P by dx.^, d\\, dc-^^. Then those of 'P^ will be 

 dxi -\- vdt^, (/a'i, dc-^, and those of 0., will be v (dt-^ -\- dt.^), 0, 0. 

 Therefore 



dt. 



and 



^ , /3t' , 9t' \ , 9l' 9t' ^ 9t' ^ 



\9t 9.V / - 9x 9y 9z 



^This paragraph answers a criticism of the relativity principle put 

 forward by Sir Joseph Larmor. Proc. Nat. Acad. Sci. 4, 334, 1918. 



* Over a small region it is always possible to use rectangular coordinates, 

 even if space is curved, provided no singular points are present. 



