WILSON AND LEWIS. — RELATIVITY. 417 



sider the /'-line as the locus of a particle at rest, then simultaneous 

 points are those along .r' or along lines parallel to x'. Hence points 

 which are simultaneous from one point of view, are not simultaneous 

 from the other. In fact any two points through which a line of class 

 (7) can be drawn may be regarded as simultaneous by choosing this 

 (7)-line as the axis x, and the perpendicular line as the axis t. Sim- 

 ilarly any two points through which a (5)-line can be drawn may be 

 regarded as having the same spacial position; in other words any point 

 may be taken as a point at rest. 



It thus appears that the measurements of time and space are de- 

 termined only relative to some selected set of axes. Further to exhibit 

 this fact, and to determine the relations 

 which exist between the measures of 

 time and space when different sets of 

 axes are chosen, let us consider (Fig- 

 ure 15) two parallel (5)-lines in our 

 non-Euclidean plane. These lines 

 represent the loci of two particles 

 which have no relative velocity. Let 

 any set of axes of time and space be 

 drawn. The constant intervals cut off 

 by the two parallel (5)-lines from the 

 a:-axis and all lines parallel to this axis 

 represent the constant distance, as Figure 15. 



measured bv these axes, between the 



two particles at any time. The constant intervals cut off by the 

 two parallel (6)-lines on the /-axis and all lines parallel thereto repre-. 

 sent the constant interval of time as measured by these axes, which 

 must elapse between the instant when one of the particles has a certain 

 position (upon the line in which we are considering rectilinear motion 

 as taking place) and the instant when the other of the particles has 

 this same position. 



One particular choice of axes is especially simple, namely, that 

 in which the /-axis is parallel to the two (5)-lines, and the x-axis is 

 perpendicular. Relative to this assumption of axes the particles are 

 at rest. The distance between them is AB. If another set of axes 

 is drawn, the particles appear to be in motion, and the distance be- 

 tween them is taken as A' B' . If i/' denotes the angle between the 

 axes, the projection of A'B' on AB is equal to AB, 



AB = A'B' cosh ^P = ;^A=.- 



Vl — w2 



