﻿Shaft moving with Uniform Speed along its Axis. 749 



Thus 



p 2 _ f+s cot 8 . 90 s 



g*-f- S tenS> {V) 



sin a 



'^sliTS' (21) 



f=p*^ (22) 



r sin a v 7 



At this point assumption (B) is introduced, i. e. one of the 

 principal axes in the final position lies along the direction of 

 resultant velocity, and we write 



tan 8 = vlu (23) 



In the expression for the twist obtained in § 2 we have to 

 note that 6' is in the opposite sense to &>, since translation is 

 in the positive direction along z, and we introduce a negative 

 sign, writing 



Q'jl = -cov/c 2 \/l-v 2 /c 2 = r ; 



hence 



s = err= -uv/c 2 V ' l-v 2 /c 2 , . . . (24) 



and from (12) 



/ = sj\-v l l(? (25). 



Inserting these values in (20), we find 



p*/q2=l-v 2 lc*-u 2 /c 2 , (26) 



giving 



tan a = v/u Wl- v 2 /c 2 - u 2 /c 2 , . . . (27) 

 and from (23) and (27) 



sin a = sin 8 Sl — v>/c*-u*f? / s/\-v 2 jc 2 . . .(28} 



Inserting these values in (19), (21), and (22), we find 

 the values of e, p, and q. The contractions are, most con- 

 veniently, stated for the surface of a cylindrical shaft having 

 a uniform motion of translation ?> and a rotational speed u at 

 the periphery, where both velocities are measured b) the 

 fixed observer. 

 Symbol. Direction. Contraction. 



p Resultant velocity. V 1 — v 2 /c 2 — u 2 /c 2 . 



e Circumferential. v' l — v 2 /c 2 — u 2 /c 2 / \/l—v 2 /c 2 .. 



f Longitudinal. y/\ — v'/tf. 



q Normal to acceleration 1. 



and resultant velocity. 



