H. A. Bumstead — Lorentz-FitzGerald Hypothesis. 499 



transverse mass greater than the longitudinal, whereas the 

 opposite is the case with the apparent mass clue to electrical 

 charges. A closer consideration however shows that this is an 

 error arising from the application of the ideas of rigid dynam- 

 ics to a body which is changing its shape. 



The path of any particle of the bar, if measured by a scale 

 carried along with the earth, will appear to be the circle AB'C ; 

 if measured with reference to a scale at rest however it will 

 be the ellipse ABC in which OB = VT^ O A. For brevity 

 we shall refer to these as the ki apparent" and the "true" 

 paths. In case (1), let P x be the true position of the par- 

 ticle, P/ its apparent position; let OH, — x x \ ^L 1 F 1 = y x ; 

 < AOP, = X ; < AOP/ = 0/.' In case (2), let OM 2 = x, ; 

 M 9 P a = y, ; < BOP 2 = 2 ; < BOP/ = % '. ' The potential 

 energy of the twisted wire in either case depends on the 

 apparent angle 0/ or 0„'. This is seen if we consider two 

 pointers attached to the wire, one along OA when the wire is 

 untwisted and the other along OB ; if the wire is now 

 given any twist the two apparent angles 0/ and 2 ' will be the 

 same, but the real angles 1 and 6 2 will be different as well as 

 the two elliptical arcs traced out by the ends of the pointers. 

 As the apparent motion is isochronous we may put the poten- 

 tial energy equal to \ It 6'\ 



In position (1) we have 



• Vl = x x Vl - ^ tan 6\ 



For small oscillations, x 2 = a ; tan 0\ — 0\ and 



Thus the potential energy is - /_ R2 y* ; the equation of 

 motion of the particle becomes 



T >\'j, = ~ 

 and the period of oscillation 



In case (2) 

 a? a = a sin 6' 2 = aB\ for small oscillations. The potential 



1 * ■ 



energy is thus -~ — a? 2 , and the period 



— a 



a; 



(1- 

 h 



■?)** 



a 2 



(1 " 



-?)"' 



>m 



, (1- 



-/}') a' 



27TJ/ 



mjj; 



