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XLI. Ihe Problem of the Uniform Rotation of a Circular 

 Cylinder in its Connexion witli the Principle of Relativity. 

 By W, F. G. Swaun, D.Sc, A.R.C.S., Assistant Lecturer 

 in Physics at the University of Sheffield *. 



EHRENTEST (Phys. Zeit.Nov. 1909) considers the problem 

 o£ the uniform rotation of a circular cylinder in its relation 

 to the principle of Relativity. He remarks that according 

 to that principle, each element of the circumference moving 



with linear velocity v should contract in the ratio ( 1— — 2 J , 



C being the velocity of light, while since each element of the 

 radius is moving in a direction perpendicular to its length, 

 the radius should not alter in dimensions. He points out 

 that the co-existence of these two conditions is impossible. 

 In the Philosophical Magazine (ser. 6, vol. xx. no. 115, 

 p. 92), Stead and Donaldson have suggested that the 

 explanation is to be found, at any rate in the case of a thin 

 disk, by assuming the disk to become cup-shaped when 

 rotating, and they have calculated the shape of the cup as 

 a function of the velocity of rotation which is necessary to 

 ensure non-violation of the above conditions. They suggest, 

 moreover, that in the case of a cylinder of appreciable 

 length, the cylinder more probably becomes strained. It 

 must be observed, however, that the assumption of the cup- 

 shaped body involves a displacement of the particles of the 

 disk in a direction perpendicular to its initial plane, the 

 relative displacement of the particles varying with the 

 distance from the axis of rotation ; and it seems that this 

 displacement of the particles would be as inconsistent with 

 the assumed principle of relativity as a contraction of the 

 radius would be. Further, suppose that instead of con- 

 sidering a flat disk, we consider a disk which when not 



Fig. 1. 



L 



A £ 



rotating has a shape the section of which is shown in 



nV 1, A. If this disk is set in rotation, and there is to be 



* Communicated by the Author. 



