Parabolic Trajectory on the Theory of Relativity. 677 



and ct for each d> and plotting against x (shown on the other 

 curves). The points on these curves corresponding to even 

 valnes of v/c and ct could then be projected on the path. 



Fig. 1. 



•80 X 



10 



It will he noticed that, the graph for ct against oc is nearly 

 straight, showing that the equable description of horizontal 

 distance is not seriously departed from even at the high 

 velocities taken. The analytical reason for this is seen on 

 inspection of the formula? (6) and (8) for x and ct. It will 

 be seen that the expressions are built up of the same two 

 functions of $ with different constant multipliers. The first 

 term in ct is cosec a times the term in x\ whereas in the 

 second term the ratio is sin a. The two functional expressions 

 are roughly of the same order of magnitude in ,/• for the 

 smaller values of (£. but. as the limiting value i< approached, 

 the second term becomes insignificant compared with 'the 

 first. Evidently the ratio of ct to ./' is mainly determined 

 by the first term and approaches the constant value cbsec *. 



