322 Messrs. H. Donaldson and G. Stead on 



energy of the disk. Let us consider, first of all, the increase 

 of mass. To calculate this increase we have to make some 

 assumption, and we shall adopt as our hypothesis the most 

 general principle of mechanics, namely, the conservation of 

 linear and angular momentum. In our particular case this 

 means that the angular momentum of the moving disk, 

 relative to the fixed disk, as measured by an observer on the 

 fixed disk will have the same numerical value as the angular 

 momentum of the fixed disk relative to the moving disk as 

 measured by an observer on the moving disk. 



Now, angular momentum is given by terms of the form 

 mr 2 co, and r is measured in the direction perpendicular to 

 the direction of motion of the ring element considered, and 

 is therefore the same on both systems of units. This leaves 

 us with the fact that 



m 

 m Q 



= 



CO 



= 



t 



where the letters without suffixes denote units on the moving 

 system and those with the zero suffix denote the units of the 

 same quantities on the fixed system. 



Now, we have shown that the time unit on the moving 

 system is greater than the time unit on the fixed system in 

 the ratio (1 — v 2 /c 2 )~* : 1, and therefore we have 



m — m ( 1 — v 2 /c 2 ) ~ * , 



which is the formula previously deduced by oiher writers 

 for uniform motion, and experimentally confirmed for such 

 motion by the researches of Kauffmann and Bucherer on 

 moving corpuscles. If now we assume, as a second 

 fundamental principle, the constancy of Total Energy of our 

 moving system ; that is, that the Total Energy of the con- 

 tracted disk is the same as that of an uncontracted disk 

 moving with the same angular velocity, we can calculate an 

 expression for the change in the internal energy of the disk 

 due to dimensional and mass changes which have occurred 

 in it on the relativity theory. 



We have that the kinetic energy of the disk, corresponding 

 to an angular velocity a), is 



E=4Ia> 2 , 

 and therefore 



d'E = Icodco-\-^(o 2 dJ. 



If we consider our curved disk we shall be able to regard 

 o) as on the fixed system of units and shall be able to proceed 



