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LXXVIII. The Modification of the Parabolic Trajectory on 

 the Theory of Relativity. By W. B. Morton, M.A., Queen's 



University, Belfast*. 



THE simplest case of accelerated motion, that in which 

 a particle moves in a straight line with constant i{ rest- 

 acceleration/ 5 was investigated by Bornj" who gave it the 

 name of " hyperbolic motion '' because the distance-time 

 graph, or " world-line/' has that form. The next in order 

 of simplicity is that which corresponds to the parabolic path 

 under gravity — i. e., when the particle has rest-acceleration 

 constant in magnitude and direction, but possesses a velocity- 

 component perpendicular to this direction. 



Let the origin be taken at the point where the velocity 

 is perpendicular to the acceleration. It will be convenient 

 to speak of the tangent to the path at this point as " hori- 

 zontal " and the direction of the acceleration as the down- 

 wards vertical. Let the axes of x and y be taken in these 

 directions respectively, u is the velocity at the origin, v the 

 velocity at any point, and c the speed of light. The in- 

 clination of the tangent to the horizontal will be denoted 



The rest-acceleration is derived from the ordinary 

 acceleration as estimated from a "fixed" standpoint by 

 multiplying the tangential component by y d and the normal 



/ 2\ — 



component by y 2 where 7=l/( 1 2 ) • 



. vdv 

 Therefore in the present case the resultant of y z —j- 



v 2 ... 



along the tangent and 7 s — along the normal is in the direc- 

 tion of Oy and has the constant magnitude / say. The 

 equations of motion are 



vdv v 2 



7 3 — t-cos<£— 7 2 — sin = . . . . (1) 



v 2 

 and 7 2 - =fcosd> (2) 



P ' 

 dv 

 Equation (1) gives y-ji =vtan<£. 



Put v = c sin 6 so that 7 = sec 6, then cosec 0d0 = tan <j)d<fi, 



giving the relation 



tan 10 . cos = const. = tan ±0 Qi say, . . (3) 



where sin6 = u/c. 



* Communicated by the Author. 



f Born, Ann. d. Phys. xxx. p. 1 (1909). 



