320 Messrs. H. Donaldson and G. Stead on 



curved space." This cuplike representation of our disk is 

 a method of representing the contracted disk in ordinary 

 space, and the results obtained by considering it will still 

 hold good, even if the disk really remain plane. 



Let us now consider an observer on a fixed disk regarding 

 the moving disk. The number of revolutions performed by 

 the moving disk relative to the fixed disk as measured by the 

 observer on the fixed disk will obviously be the same as 

 the number of revolutions of the fixed disk relative to the 

 moving disk as measured in the same period of time by an 

 observer supposed placed on the moving disk. That this 

 will be so follows from the fact that a number is of no 

 dimensions in length, mass, or time. But to perform one 

 revolution a point on the moving disk has to traverse a 

 linear distance which is less than that traversed by the 

 corresponding point of the fixed disk in the ratio 



Hence, in order that the two numerical measures should 

 be the same, it is necessary that the time unit on the moving 

 system should be greater than that on the fixed system in the 

 ratio (1— v 2 /e 2 )~* : 1. This is the result which has previously 

 been deduced by other writers in a variety of ways for 

 systems in linear motion. 



We shall now proceed to calculate the kinetic energy of 

 our rotating disk, and to show how this leads us to the 

 necessity for a change in the mass of the system due to its 

 rotation. 



Since a ring element of area 2irs . 8s on the fixed disk 

 becomes a ring element of area 2iry . 8s on the moving disk, 

 and since 



y = S{l-V 2 /C 2 )i, 

 we shall have 



o' = <r (l-v 2 /c 2 y% 



where a and a are the surface densities of the fixed and 

 moving disks respectively. 



Now v = yco, the units being those of the fixed system, 

 and therefore, since 



we have 



