General Relativity and Einstein's Theory. 391 



li S and S' are reciprocal Euclidean systems characterized by 

 a constant light velocity c, such that every point in S' has the 

 same constant velocity v relative to S, 



C' = C = c, 



v' = - V, 



l1/^-? 



and integration of equations (5), (6), (7) and (8) gives the 

 Lorentz-Einstein transformations of the restricted principle of 

 relativity. 



Put l=tct, 



C 



Then if S' is a Euclidean system characterized by the constant 

 light velocity c, (9) becomes 



dr'^- -\- dl'^ = Ir (dr-^n^dP) 



=h-dr- + k-dP, (11) 



in which greater symmetry has been obtained by replacing the 

 variable t by /. Put 



ds^ = dr'^ + dl'\ 



As S' is a Euclidean system points may be located in it by 

 means of rectangular coordinates x', y', z'. Therefore, if x', y' , 

 z' and /' are interpreted as the rectangular coordinates of a point 

 in a four dimensional representative space, every moving element 

 describes a path through this space, of which ds is an element 

 of arc. Minkowski was the first to use this representation, and 

 the path of the moving element is called by him its ivorld-line. 

 It has been noted that a physical phenomenon is a coincidence in 

 space and time of two or more moving elements. Such a coinci- 

 dence is represented by the intersection of the world-lines of the 

 moving elements concerned. Therefore all physical measure- 

 ments are confined to the observation of such intersections, and 

 all representations which make these intersections occur in the 

 right order are equally valid in describing the results of experi- 

 mental science. 



