676 Prof. W. B. Morton on the Modification of the 



through a right angle and approaching coincidence with 

 the acceleration, as in the parabolic case, is turned through 

 / \ 



( - —a) and the particle approaches a steady motion at the 



speed of liglit in a direction inclined at angle a to that of 

 the constant " rest-acceleration/' The following table gives 

 the values of a for different values of u/c : — 



ujc. a. 



•1 2° 52' 



•2 5° 48' 



•3 8° 50' 



•4 12° 2' 



•5 15° 33' 



•6 19° 28' 



•7 24° 6' 



•8 30° 0' 



•9 38° 49' 



1-0 . 90° 



For u/c — *8, sina = {c — (c 2 — u 2 )^}/u=^. 



To gain a clearer view of the terminal conditions, which 

 have a rather paradoxical appearance at first sight, it is 

 worth while to evaluate the components of the (ordinary) 

 acceleration. We find, reinstating/, 



vdv/ds=f sin </>(cos 2 cp — sin 2 «) 3 /(cos 2 </> 4- sin 2 a) 3 , 



v 2 /p=f cos </>(cos 2 <j> — sin 2 a) 2 /(cos 2 <£ + sin 2 a) 2 , 



and ry=( CO s 2 $-4- sin 2 a) /(cos 2 <j> — sin 2 a), 



so that y 3 times the former component and <y 2 times the latter 

 combine to give/. This verifies the analysis. 

 Now put <£ = 7r — a — % where y, is small, 



vdv/ds=f cos a . cot 3 « . % 3 , 



V 2 



~ =/'sin cc . cot 2 a . y 2 

 p J A 



7 = tan ot/x- 



The tangential. acceleration (ordinary) is small compared 

 with the normal, but in passing to the rest-acceleration it is 

 multiplied by a higher power of the large quantity 7, so that 

 the modified components are of the same order of magnitude 

 and have the finite ratio tan a. 



I have taken the case u/c = '$, a = 30°, and have calculated 

 x, y for different values of <f> and plotted the path shown on 

 the diagram. The parabola with the same " a " is shown 

 for comparison. The figures marked above the path give 

 the values of v/c at the corresponding points, and those 

 below give the time, the unit in the latter case being the 

 time required for light to describe the parameter " «." 

 These points were obtained by calculating the values of v/c 



