the Problem of Uniform Rotation. 321 



The moment of inertia of the disk about its axis of rotation 

 can now be obtained immediately, for, taking our ring- 

 element, we have 



and y = s ( 1-f — §~ J ' 



|_2&> 2 2&> 4 °V c 2 /J. 



When o) is small this reduces, as a first approximation, to 

 the ordinary expression for the moment of inertia of a 

 circle about an axis through its centre perpendicular to its 

 plane. 



The kinetic energy of the disk is given by |Io> 2 , that 

 is, by 



irar 2 c 2 c* , /. r 2 co 2 \ 



-2 — »?M 1+ ~?"/ 



When (o is infinite this becomes 



i Mc 2 - SrLta^co ■. 



2 o)" 



The limit term is seen, in the ordinary way, to be zero, 

 so that the maximum kinetic energy possible for the disk 

 is pic 2 . 



Now, when the disk possesses this energy its volume has 

 become zero, as shown in the preceding paper, and therefore 

 we arrive at the seeming absurdity that a finite expenditure 

 of energy is able to reduce the volume of the disk to zero. 



This conclusion can be avoided in two ways, or by a 

 combination of the two. The possibilities are, first, an 

 increase in the mass of the moving disk, due to its velocity, 

 and, secondly, an increase in the internal (" potential ,? ) 



