The Second Law of Thermodynamics. 575 



which these quantities are t, i\. . . v r , to a varied motion, in 

 which they are T + "dr, v^B^i, &c, we have (1) to supply 

 energy dE, (2) to do work XF'dv against the constraining 

 forces. The whole energy required is dE + 2P^v, or 



*+*M£-£)*- 



5. Now let there be a great number, N, of such molecules, 



all with the same values of v x . . . v r , but different values of t 



and E. According to the provisional assumption above made, 



the N molecules are describing periodic motions with different 



periods, and with an infinite variety of phases. Owing to 



dV 

 this variety of phases -7- = for each V, not only on average 



of the time, but on average at any instant of all the molecules 

 describing a given orbit. And if the whole motion be sta- 

 tionary, this property may hold even though the motions of 

 individual molecules be not strictly periodic. We may then 

 replace the provisional assumption, that each molecule is in 



dV 

 periodic motion, by the assumption that >-=- = for each v. 



6. Let S/(#i . . . x n ) daji . . . dx n , or, shortly, "Nfda; be the 

 number of molecules whose coordinates at any instant lie 

 between the limits 



x 1 and a&i + dj&u^ 



x 2 and x 2 + dx 2j > (A) 



&c. J 



Then fda is the chance that any given molecule at any instant 

 shall belong to that class. 



Similarly let f(oci . . * Xn)dxi . . . dx n , or, shortly, fda', be 

 the chance that for a given molecule at any instant the velo- 

 cities shall lie between the limits 



d-i and ki + dx ly ~>> 



x 2 and x 2 + dx 2 , r ffi) 



&c. J 



Whether or not f be a function of the coordinates a t . . . x n , 

 we must have 



The chance that a given molecule shall at any instant belong 

 to both class A and class B is ff da da', or, as we will 

 write it, F da da' . 



2R2 



