ON THE KINETIC THEORY OF GASES. 207 



of an infinite number of such molecules sparsely scattered, 

 the chance that any given one shall have co-ordinates be- 

 tween the limits 



q x and q 1 + dq y \ 



q 2 and q 2 4- dq 2 > (a) 



etc., ) 



and momenta between the limits 



p 1 and p t + dp z \ 



p 2 and p 2 + dp 2 \ (b) 



etc., ) 



in proportional to 



■ -*<*+*> fl^ x . . . dp n 

 where ^ ls the potential, T the kinetic energy in the state 

 (ad). This is the general law of permanence of which the 

 £-' l, ""' 2 du of elastic spheres is a particular case. 



10. We must consider the assumptions on which this 

 theorem rests. Let f(q 1 . . . p n ) dq t . . . dp n be the 

 number of molecules in unit of volume between the limits 

 "a" " <£ " above, or the chance that a given molecule shall 

 be between those limits, and let F (Q x . . . P„) dQ l . . . dP tl 

 be the number between the corresponding limits 



O x and Q r + dQt \ 



Q 2 and Q 2 + dQ 2 i A 



etc., ) 



and 



P, and P r + d?, \ 



P 2 and P 2 + dV 2 \ B 



etc., ) 



The number of pairs, one from each set, is shortly f¥dq 1 

 . . . dP n . Let these values of the variables denote 

 the state in which mutual action between the pair com- 

 mences. Such a pair will in time r pass by their own 

 mutual actions, uninfluenced by any third molecule, into 

 the state 



and 



(a) 



(/>') 



