254-257] Calculation of Subsidiary Temperatures 213 



and similar equations for the other suffixes. If we replace 0^ by its value as 

 given by equation (501), we obtain 



(510), 



and these equations lead to equations which are formally the same as 

 equations (503), namely 



r 1 = l f 1 (T\ etc. 



except that/i(r) no longer stands for 



but for 



The value of fi(T), it will be noticed, is as before, independent of v, 

 for s lt Si are each directly proportional to v. 



256. We have now seen that the values of the subsidiary temperatures 

 are given by equations of precisely the same form, whether we suppose 

 (i) that all the energy which passes into the ether is lost to the gas, or 

 (ii) that all the energy is reabsorbed by the gas. It is therefore pretty 

 obvious that the equations will also give a fair account of the state of things 

 which will arise when part only of the radiated energy is reabsorbed by the 

 gas. We shall, accordingly, in the present chapter, investigate the behaviour 

 of a gas upon the supposition that the subsidiary temperatures are given by 

 equations of the form of equations (503). 



257. Before passing on it may be well to remark that there is a single 

 case of failure of these equations. This occurs when X 1 = 0, and is not 

 equal to zero, for then the value of TJ. given by equation (510) becomes 

 infinite, suggesting that, in arriving at this equation, it is no longer 

 legitimate to regard r-i as small. The physical interpretation of this case is, 

 however, obvious. The combined energies E l and r a can continue to gain 

 energy through collision at the expense of T, but have no opportunity of 

 losing energy, until r x becomes comparable with T. When this stage is 

 reached energy can be repaid through the medium of collisions. The final 

 stage is one in which the exchanges of energy produced by collisions are 

 exactly balanced. This will be seen to be a state in which the material 

 energy is equally divided between the different degrees of freedom, while the 

 energy of the ether stands in a definite ratio to the material energy, the 

 absorption exactly balancing the radiation. 



This state, however, is not likely to occur in nature. It requires in the 

 first place that the energy of the ether shall produce absolutely no effect on 

 the translational energy of the molecules, and in the second place that the 



