324 On the Problem of Uniform Rotation. 



Now 



from our above expression for I, and, to a first approximation, 

 this becomes 





E= ir r i7 



But 



E =JirrW, 



and therefore 







E -E_ j^^o-^- 





1 ©V 

 " 3 ~?~ Bo ' 



Hence, to a first approximation, the change in internal 

 energy due to the contraction of the disk is given by 



1 M 2 T 2 



P-P =-^E, 



In conclusion we consider ihat this investigation shows 

 that the relativity theory involves no contradictions when 

 applied to the case of uniform rotation, and also that the 

 case of uniform rotation gives a very sound and direct 

 method of proceeding from length to time units, depending 

 only on the fact of a number having no dimensions in mass, 

 length, or time. Again, the maximum kinetic energy 

 possible for the disk shows the necessity for some change in 

 mass, though it does not give, of course, any result which is 

 uniquely satisfied by the relativity change of mass-unit. It 

 will be noticed that we have given no proof of the change in 

 the length-unit of the moving system, the reason tor the 

 omission being that the change can be directly deduced on 

 the method given by Messrs. Lewis and Tolman (Phil. Mag. 

 Oct. 1909) for uniform rectilinear motion. 



Cambridge. 

 October 31, 1910. 



