Probabilities to the Movement of Gas-Molecules. 263 



different Q's and q's, transform T so as to form a linear 

 function of squares, say 



c 1 n 1 2 + an 2 2 + ...c l m n wt 2 +r ] 7r 1 2 +...r, i 7r / r ; . (26) 



where the coefficients are numerical. Then, as in the 

 simpler case, jT must be of the form 



iCilTi 3 + . . . 4- iC m TIm 2 + i^TTi 2 + . . . + iC n ir n 2 , 



1 C 1 = C 1 (m + n)/2[T], ... Cll = lCl (m + n)/2[T], ... 



the condition analogous to (4) will be satisfied. Also as 

 1 T = 1 T' (the mean energy of the set which has come into 

 collision during At) the other equations for the constants 

 must be satisfied (see (13) and (34)). Thus the distribution 

 may be written 



Z = Jexp-i'^|-T/[T], . . . (27) 



The complex molecules with different co-ordinates to 

 which this distribution refers form a ; ' universe," which 

 may be divided into genera delimited by values of the co- 

 ordinates. Consider first extensive genera, e. g. all the 

 cases which have co-ordinates between Q x and Qx + a, 

 Q 2 and Q2 + /3, where a, /3, etc. are considerable. If the 

 normal law is very ivell fulfilled for the " universe " it must 

 be fairly well fulfilled by those genera*. As the law 

 becomes still better fulfilled for the universe it must become 

 very well fulfilled for those genera. So by parity of reason 

 it must be fairly well fulfilled for subordinate less extensive 

 genera ; and so on. Ultimately the law must be fulfilled 

 for every genus denned by neighbouring values of the 



co-ordinates Q x and Qi + AQi, Q2 and Q 2 + AQ 2 The 



mean energy for each such genus is presumably the same f. 



II. The second argument gives at once the distribution in 

 terms of the co-ordinates, with reference to each particular 

 configuration which may be assumed by a pair of colliding 

 molecules. Let T be the energy of a pair expressed in 

 terms of the co-ordinates, and either the generalized velo- 

 cities or the components of momentum ; and let P T be the 

 momentum in the direction of a fixed axis x, and likewise 



* The constituents of the universe being taken at random from those 

 genera, supposed few. 



t The energy would be the same for any one molecule moving without 

 collision through different phases denned by the varying values of the 

 co-ordinates ; and there is no reason why it should not be the same for 

 the average molecule in different phases. 



