774 



Einstein s Law for Addition of Velocities. 



Two additional points may be mentioned. First: — Seeing 

 that the curves are convex upwards in fig. 1 and downwards 

 in fig. 2, it is evident that they may become approximately 

 straight for an intermediate value or! 7. The tangent to the 

 curve at the axis of y is directed towards the terminal point 

 (1 1) for sec y=l-\-u. In tliat case there cannot be much 

 variation from the straight line — i. <?., the speed increases in a 

 linear manner up to that of light as v is given larger and 

 larger values. This is seen on the diagram (fig. 3) which is 



•5 i-o 



for ry = 45° ; there is not much deviation from straightness in 

 the graphs on the positive side o£ the origin. 



Second : — It comes out that the condition for a resultant 

 of minimum size, when uy are given, is identical with that 

 found on the old theory — viz., v= — u cosy. The square of 

 the resultant may be written in the form 



1 (t-„«)(i-^ 



(1 -f uv cos y) 2 ' 



and (1 — v 2 ) j(l + uv cosy) 2 is maximum for v = — i^cos^. 



To show the continuity of passage from fig. 2, through 

 fig. 3 to fig. 1, the parts of the curves in fig. 1 which lie 

 below the axis must be reflected upwards so as to give the 

 magnitude of the resuliant without respect to its direction. 



