772 Prof . W. B. Morton on Einstein's La 



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values being given by the intersections of the curves with 

 the axis of y. 



The curves are portions of rectangular hyperbolas having 

 centres at x — — lfu, y = lju, and asymptotes parallel to the 

 axes of xy. They run to the terminal points which corre- 

 spond to the velocity of light in the forward and backward 

 directions. On the old theory the graphs would, of course, 

 be straight lines, starting from the same points on the y-axis 

 and making 45° with the axes. 



The most interesting features of the curves are in the 

 region of negative v (backward motion along the platform). 

 The resultant vanishes for v= — u, just as in the old theory, 

 although now the " equal " speeds are measured with different 

 standards. This is merely another way of expressing the 

 reciprocal nature of the Lorentz transformation : the ground 

 appears to move backward with velocity u from the platform 

 standpoint. When v— — 1 the resultant has the same value ; 

 a light-signal travelling backward has unit velocity to both 

 sets of observers. 



When u is large there is a very sudden increase in the 

 resultant speed as v passes from — u to — 1. For example, 

 if the platform moves at *9 of the speed of light, a point 

 moving backwards along it at this same speed will appear 

 stationary from the ground. But if the speed is pushed up 

 to the full value for a light-signal, this has its full value also 

 as seen from the ground. On the diagram this is shown by 

 the sudden downward plunge of the curve for *8. As the 

 value of u is increased towards unity the hyperbola ap- 

 proaches the' rectangular lines along the top and down the 

 left-hand side of the diagram. 



From this consequence of Einstein's formula there follows 

 a curious paradox which I have not seen mentioned, and 

 which may be added to the many odd things which would 

 happen if anything but light could be made to move as fast 

 as light. 



First let the platform have any speed, and let a point move 

 backwards along it with the same speed as measured on the 

 platform. Then the point is at rest as seen from the ground. 

 Increase the common speed up to that of light and we have 

 a light-signal moving backward along a platform which 

 moves forward at the speed of light and appearing stationary 

 from the ground. 



But now let the backward-moving thing on the platform 

 be a light-signal from the first. This, on the fundamental 

 assumption of the theory, moves with the velocity of light as 



