﻿746 Prof. Hackett on Relativity- Contraction in a Rotating 



of the Wiedemann effect the shaft was taken as a thin tube; 

 this restriction is not necessary for the general mathematical 

 treatment, as can be seen by reference to deduction (I.), 

 though it may be helpful to think of the shaft as a thin tube 

 in the following discussion. 



We consider that the new co-ordinates x\ y', z' of any point, 

 as interpreted by the fixed observer, are given in terms of 

 the original co-ordinates as in the ordinary strain theory. 

 The axis of the shaft is taken as the axis of z. The fixed 

 observer infers a rotation 6' — tz according to (3), and a 

 FitzGerald-Lorentz contraction parallel to z according to 

 deduction (II.). In accordance with assumption (A), each 

 radius in any plane parallel to (x, y) is assumed to be con- 

 tracted in a ratio which depends only on r, and the circum- 

 ference alters in the same ratio. 



As the rotation 6' cannot be assumed generally to be 

 small, the steps of deducing the strain-components are 

 given below : — 



x 1 = ex cos 6' — ey sin 0\ 



y' = ex sin 6' + ey cos 6' . 



We get for the relative displacement £/, 7} t ', £*' around 

 x, y, z, of a point whose undisplaced co-ordinates are x-\-^ y 



y + tlyZ+K- 



But part of these relative displacements is a pure rota- 

 tion 6' around an axis through x', //, z' parallel to the axis of 

 the cylinder and arising from the general rotation. We shall 

 obtain the strain components f, V, f by combining the total 

 effect £/, rjt, ?/ with a rotation — 0', where 



f':= f/ cos d' + v/ sin 0' , 



v ' =■-.&' sin 0' + V cos 0'. 



Whence 





