104 The Law of Distribution [CH. v 



the case of the loaded sphere, let us suppose almost the whole mass concen- 

 trated at a point which must of course coincide very nearly but not quite 

 with the geometrical centre, so that the weight of the remainder of the 

 molecule is very small by comparison. Then k is very small, and since 

 A;^! 2 is on the average equal to c 2 independently of the value of k, it 

 follows that CTJ must be very great, becoming infinite in the limit when 

 k vanishes. In this case, then, the molecules will rotate with infinite 

 angular velocity. 



117. If the values of H and K are slightly disturbed from the value 

 appropriate to the steady state, we have, from equation (254), 



shewing that the disturbance will decrease exponentially with the time, and 



will fall to - of its value in a time 

 e 



3 



of which the value, on replacing /3 from equation (251), and putting 



3 

 K = 7T from equation (161), is found to be 



\., ...(256). 



The factor in brackets is the mean duration of a free path (cf. equation 

 (48)), so that the time in question is 



x (the average duration of a free path) ............ (257). 



This quantity naturally depends both upon the structure of the molecules 

 and the state of the gas. Its reciprocal will, so to speak, measure 'the 

 amount of " grip " which the translational motion is able to obtain over the 

 rotational. 



The problem of partition of energy in a system of loaded spheres was 

 st investigated by Burnside*. His result was, however, erroneous. The 

 correct result was subsequently obtained by Burburyf-. In the same paper 

 Burbury calculates the rate of subsidence of a disturbance in a gas con- 

 sisting of two kinds of symmetrical molecules, the disturbance consisting of 

 a small inequality in the mean translational energy of the two kinds of 

 molecule. The corresponding calculation for loaded spheres was, so far as 

 I know, first given in papers by myself]:. 



* "On the Partition of Energy between the Translatory and Botatory Motions of a set of non- 

 homogeneous Elastic Spheres," Tram. R. S. E. xxxm. Part ii. (1887). 



t "On the Collision of Elastic Bodies," Phil Trans. CLXXXIII. p. 407 (1892). 

 J " The Distribution of Molecular Energy," Phil. Trans, cxcvi. p. 397 (1901) : and "On the 

 Partition of Energy in a system of Loaded Spheres," Quarterly Journal, xxxv. p. 224 (1904). 



