Oil Uniform Rotation, §c. 821 



is not a matter for surprise that any attempt to extend the 

 scope of the principle into this domain leads to immediate 

 difficulties. Ehrenfest* has propounded the paradox that 

 if a disk is spinning uniformly, any circle of a concentric 

 system must contract in a definite ratio defined by the velocity 

 of one of its points, for each element of its arc is moving 

 along the tangent with a uniform velocity. Yet at the same 

 time, the radius of the circle, which has no motion along its 

 length, should not alter. An ingenious mode of reconciling 

 these statements has been devised by Stead and Donaldson t, 

 who suggest that the rotation causes the disk to buckle 

 inwards, taking the form of an epicycloidal surface of revo- 

 lution when the rotation is small. They also regard it as 

 probable that a solid cylinder would be more likely to remain 

 in a state of strain, it appears that the requirements of the 

 principle of relativity can be satisfied in this way, although 

 it might perhaps be preferable, in place of the somewhat 

 unsatisfactory hypothesis of straining, to suppose that the 

 cylinder became distorted so that normal circular sections 

 became parallel surfaces, epicycloidal for small rotation 

 about the axis. 



There are many other difficulties connected with uniform 

 rotation. For example, if a disk of sufficient radius were 

 set into rotation, the particles at a sufficient distance from 

 the axis would acquire a velocity greater than that of light, 

 as Mr. A. A. Robb suggested to me in the course of a dis- 

 cussion. Quite apart from the principle of relativity, there- 

 fore, if it is possible to set a disk of sufficient size into 

 rotation, the only way of representing the phenomena which 

 must occur, by the medium of ordinary geometry, appears 

 to be by a supposition of a buckling effect, in default of an 

 indefinite contraction of the radius. 



The essential distinction between uniform rotation and 

 uniform translation is, from one point of view, the presence 

 of a normal acceleration in the former, and this must render 

 the principle of relativity, at any rate in its ordinary 

 form, inconsistent with the electromagnetic equations. But 

 this normal acceleration is proportional to the square of 

 the angular velocity, and therefore cannot introduce more 

 than a second order effect into the general nature of the 

 scheme of vectorial relations appropriate to translational 

 motion, for distances from the axis whose product with the 

 angular velocity is small in comparison with the velocity of 



* Phys. Zeit. November 1909. 



t Phil. Mag. July 1910; March 1911, 



