223-227] Subsidiary Temperatures 189 



momentoids. In the case of a conservative gas, it was shewn that if mu*, 



.. denoted the mean values of raw 2 , c^ 2 ... averaged over all the 

 molecules of the gas, then (equation (219)) 



raw 2 = raw 2 = mw z = c^ 2 = . . . = RT. 



In the present case, we could, if our law of distribution were known, 

 calculate the values of raw 2 , c^i 2 , etc. It is obvious that we should have 

 raw 2 = raw 2 = raw 2 , and we may as before put each quantity equal to RT, these 

 equations defining T. We shall also put c^ = Rr 1} c#)J = Rr^ . . . , etc., 

 these equations defining T I} r 2 .... It will be convenient now to speak of T 

 as the principal temperature, and of r 1} r 2 ... as subsidiary temperatures. 

 The temperatures are accordingly defined by 



raw 2 raw 2 Oi 2 c 2 _ p 



T " T " T T! " r 2 



226. The information which has been obtained as to the law of distri- 

 bution of coordinates admits of easy interpretation in terms of principal and 

 subsidiary temperatures. The result is as follows. If we start an infinite 

 number of gases, each with energy E , these gases being distributed with 

 uniform density throughout the part of the generalised space which corre- 

 sponds to energy E , then by the time the energy has been reduced to E l9 

 all except an infinitesimal fraction of the whole will have the same principal 

 and subsidiary temperatures. 



227. We can conveniently represent the temperatures of a gas in a 

 generalised space, the coordinates in this space being T, r l} r 2 Obviously 

 only those parts of the space are needed in which every coordinate is positive. 

 The motion of a gas can be represented by the motion of a point in this 

 generalised space. Our theorem now shews that of an infinite number of 

 representative points which start from the same point, all except an infini- 

 tesimal fraction pursue the same course. 



Of the representative points for the system of gases, starting with any 

 energy E , the systems being as we have assumed throughout equally distri- 

 buted throughout the original generalised space, all except an infinitesimal 

 fraction will, it is true, start from a point in the line 



T=T 1 = r 2 = ................................ .(458), 



on which there is equipartition of energy. It is however obvious that if we 

 had considered only those systems of which the representative points started 

 from the most general point (T, T I} r z ...), the result of the last chapter 

 could have been obtained in exactly the same way. It will therefore be seen 

 that the whole space can be mapped out into stream lines, in such a way 

 that when a representative point is once moving on a stream line, it is 



