Parabolic Trajectory and Relativity. 675 



When is small this equation obviously reduces to 

 v cos </> = const, in accordance with the elementary theory. 



The subsequent analysis is made much neater by the 

 introduction of an angular parameter a defined by 

 sin a=tan ^0 ={c— (c 2 — u 2 )*} fu. 



Using this equation (3) gives the connexion between 

 magnitude and direction of v in the form 



r/r = sin# = 2 sin a cos </>/(cos 2 (/) + sin 2 a). . . (4) 



From equation (2) p — c' 2 tan 2 6 sec cp/f. 



The linear scale of the path may he expressed in terms of 

 the parameter of the parabola which would be described on 

 the old theory. 



a=w 2 /2/=2c 2 sin 2 a//(l + sin 2 «) 2 . 



Using this, and the relation (3), we express the radius of 

 curvature in terms of <p and the constants a, a, 



p = ds/dcf> = 2a(l-\-sm 2 a) 2 COS <^/(cos 2 (/> — sin 2 a) 2 . 



The integral of this is the intrinsic equation of the path 



(l + sin 2 a) 2 ( 2 cos a sin </> , cos a 4- sin c/> | 



\ COS (<£-*- a) COS (</>— a) °COSa — sin <p ) ' 



•> 



COS" a 



(5) 



Integrating 



d,i'ld(f) = cos cf> . ds/dcj) and dy/d(j) = am(f) .ds/dcf>, 



we obtain for the coordinates of a point on the path the 

 expressions 



= q(l + sin 2 a) 2 f sin 2cf> + 2 cos (<£-*) j 



2 cos 2 a \cos (cp + ex.) cos (0 — a) sin 2a & ' cos ($ + a) J 



v = ''(l + sin 2 a) 2 {sec(</> + a) sec (</) — a) — sec 2 a} (7) 



It only remains to find the relation between </> and the 

 time. This can be obtained from (4) b}' writing 



v = dsjdt = ds/dcj) X dcf>/dt, 

 leading to 



cdt/d<f> = a(l -h sin 2 a) 2 (cos 2 <f> + sin' 2 a) /sin «(cos 2 $ — sin 2 a) 2 , 

 the integral of which is 



a(l + sin 2 «) 2 I sin 2$ . . cos(<£ — «) -) 

 c£= ^-^ — J — - — — + tan a log y.-- — { \ . (8) 



2 Sin a COS" a I cos ^£ + a) COS ((/> — a) ° COS (</> -}- a) J 



The following properties of the motion are seen immedi- 

 ately from the equations. In the first place as cj) approaches 



the value I - — a J, .r, y, and t become infinite and v approaches 

 c. That is to say, the direction of motion, instead of swerving 



