192 Theory of a Non-Conservative Gas [CH. vni 



II. The example of the loaded spheres. 



230. We shall be able to derive a second, and somewhat important, 

 illustration of the general theory from the problem of the loaded spheres, 

 worked out in Chapter V. We there considered a gas in which the centre of 

 gravity of the molecules did not coincide with their geometrical centres, so 

 that rotations were set up by collisions. In order to make this system of 

 molecules illustrative of a non-conservative gas, we shall suppose that energy 

 is lost when these rotations take place. We may conveniently suppose that 

 every rotation, by a mechanism which there is no need to specify with pre- 

 cision, is retarded by a force proportional to the velocity of rotation. 



As before, we shall use H, K to denote the mean energies of rotation and 

 translation of a molecule. We previously found a value for dx/dt in terms of 

 K and H, independently of the law of distribution of the angular velocities, 

 and as the change in K is still caused entirely by collisions, we can legiti- 

 mately use this expression for the value of d^jdt in the present instance. 

 The value of H decreases for two reasons. There is first a decrease of H 

 caused by collisions, and the rate of this decrease must be dx/dt, the rate of 

 increase of K caused by collisions, since the total energy remains unaltered by 

 collisions. The second cause of decrease in H is the system of retarding 

 forces. These may be supposed to cause a decrease in H of amount 2eH per 

 unit time, where e is a constant. Hence, on the whole, we shall have 



dn _ CK 



= 2eH -- - 



dt dt' 



(/K 



Introducing the value of -j- from the former investigation (equation (253)), 



we have as the equations expressing the rates of change in K and H, 



-fK) ................................. (459), 



(460). 



231. Our aim is to obtain from these equations an illustration of the 

 dynamical theory of a non-conservative gas, just as we found that the same 

 equations, with the omission of the term 2eH, supplied an illustration of 

 the much simpler theory of a conservative gas. We must begin by noticing 

 that the present system possesses only one subsidiary temperature, and that 

 K and H are respectively proportional to the principal and subsidiary 

 temperatures. 



Let us write 



(461), 



