354 FINAL STATE OF A SYSTEM OP MOLECULES IN MOTION, &C. 



Since the number of the molecules does not vary during their motion, this 

 quantity ia zero, whatever the values of f, 17, . Hence we have in virtue of the 



last term 



dA dA dA . 



or A is constant throughout the whole region traversed by the molecules. 



Next, comparing the first and second terms, we find 



c = AM(2$ + B) (19). 



We thus obtain as the complete form of dN 



G&zVj c < t.i' 1 1 */ 1 iz< I c; < IT] ( ( {, (20K 



when A is an absolute constant, the same for every kind of molecule in the 

 vessel, but B l belongs to the first kind only. To determine these constants, 

 we must integrate this quantity with respect to the six variables, and equate 

 the result to the number of molecules of the first kind. We must then, by 

 integrating dN l ^M l (' + /!* + ' + 2^,) determine the whole energy of the system, 

 and equate it to the original energy. We shall thus obtain a sufficient number 

 of equations to determine the constant A, common to all the molecules, and 

 B t , B v &c. those belonging to each kind. 



The quantity A is essentially negative. Its value determines that of the 

 mean kinetic energy of all the molecules in a given place, which is f- -T- , and 



therefore, according to the kinetic theory, it also determines the temperature 

 of the medium at that place. Hence, since A u in the permanent state of the 

 system, is the same for every part of the system, it follows that the tempera- 

 ture is everywhere the same, whatever forces act upon the molecules. 



The number of molecules of the first kind in the element dxdydz, 



(21). 



The effect of the force whose potential is ^ is therefore to cause the 

 molecules of the first kind to accumulate in greater numbers in those parts of 

 the vessel towards which the force acts, and the distribution of each different 

 kind of molecules in the vessel is determined by the forces which act on them 

 in the same way as if no other molecules were present. This agrees with 

 Dalton's doctrine of the distribution of mixed gases. 



