392 Leigh Page, 



If a particle is at rest in the system S' under discussion its 

 world-line is a st,raig"ht line parallel to the L' axis. Therefore 



ds = dl', 

 and the linear element ds is a measure of the time elapsed relative 

 to the system S' in which the particle is at rest. 



If a particle has a constant velocity v' relative to S', its world- 

 line is a straight line inclined to the L' axis by an angle whose 



tangent is 



.v' 

 c 



Let 5^ be a Euclidean system reciprocal to S' . Then (ii) be- 

 comes 



ds- = dr'~ + dP = dr- -\- dP, (12) 



and vS^ can differ from S' only in the orientation of the axes. 

 Hence every point in 5" must have a constant velocity relative to 

 S', bearing out the statement previously made that if vS and S' 

 are two reciprocal Euclidean systems characterized by the light 

 velocity c, all points in one of these systems must have the same 

 constant velocity relative to the other. 



In particular consider the case where the Y and Y' axes and 

 the Z and Z' axes coincide, and the X axis inclines toward the 

 L' axis so as to make an angle a with the X' axis, where 



Then 



y' = y> 



z' = z, 



which are the Lorentz-Einstein transformations for .the case 

 where the relative motion of the two systems is in the X X' 

 direction. 



It is of interest to inquire what systems are possible in which 

 h and k are functions of x, y and z but not of /. For simplicity, 

 consideration will be confined to motion in a straight line, i. e. 



