﻿Shaft moving with Uniform Speed along its Axis. 741 



on the same principle as the ideal clock consisting of a beam 

 of light reflected between two mirrors, with the addition that 

 a disk fixed on the shaft at any cross-section and rotating 

 with it can, owing to the twist, indicate the local time there. 

 In the latter part of the paper, the contraction in the shaft 

 due to the motion of translation and the twist is considered 

 as a strain-displacement. One of the principal axes of the 

 strain is assumed to be the direction of resultant velocity 



^rN u 2 . The principal contraction in the latter direction is 

 found to be V\- ( v * + u 2 )/c\ 



This result holds for a shaft of any form, since the twist 

 and longitudinal contraction do not depend on the form of 

 the shaft. Passing from the case of a solid circular cylinder 

 to the limiting case of a disk rotating without any motion of 

 translation, the reasoning in this paper gives the circum- 

 ferential contraction as equal to that usually accepted for a 



rotating ring, viz. v'l — u 2 /c 2 , where u is the velocity at the 

 rim. It follows that the contraction in the radius is of 

 the same magnitude. 



§ 2. The Velocity of Light and a Rotating Shaft, 



Stationary System. — A rotating shaft can serve in theory 

 for the determination of the velocity of light by the following 

 modification of Fizeau's experiment. Two similar disks are 

 mounted on the shaft in planes normal to the axis separated 

 by a distance I. Each disk is perforated by a number of 

 equidistant apertures lying on a circle concentric with the 

 shaft. In the subsequent discussion we ignore ordinary 

 elastic strains, or, in other words, assume that the elastic 

 constants are infinite. 



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

 are similarly placed so that the apertures in each disk pass 

 simultaneously through the plane of (#, z) as the shaft turns. 

 We need only consider light rays travelling in this plane 

 parallel to the axis of the shaft so that they can pass through 

 an aperture in each disk for suitable speeds of rotation. 

 Let the period of the lowest of these speeds be T. For this 

 speed, light travelling through an aperture in one disk to- 

 wards the other will pass through another aperture there 

 which has just been brought into position by a rotation of 

 the shaft through an angle (/>, where <£ = angle between two 

 successive apertures in either disk. We have then 



<j>.r.T/2ir = l (1) 



