258 On the Variation of Entropy, 



DV, DV^, &c. Each of these elements is supposed to be so 

 small that r] may (he says) be regarded as sensibly constant 

 throuohout any one ot* them at a certain instant t' , But if 

 7? is a function of the phase, this property of being sensibly 

 constant throughout any one of the elements DV, must be 

 true at every instant, and not at the particular instant t' only. 

 If 7] is not a function of the phase, what is the definition of ?; 

 for a system when t) varies with the time ? 



16. Professor Gibbs then considers another instant t" , 

 which (so far as yet appears) may be either earlier or later 

 than l' , A certain group of n systems are at t' ^ all within 

 the same extension in phase DV^, and have rf for entropy. 

 Of these n systems most, but not all, are at time ^" within the 

 extension in phase DV" and have entropy j)" . The remainder 

 are at t" scattered through many other elements of extension 

 in phase, namely, DVi^^ ViN ^' ^ &c., and have entropies t;/^ 

 t)^' ^ &c. respectively. Let us say that the number wdiich at 

 t" are in DV^^ is '^e^" , the number which are in DV/^ is 

 N^^i", the number which are in DVg^^ is Ne^'s" and so on. 

 Then since the number of the group is not altered we have 



Ne'j' = - (eV + ^m" + e'52" + &c.) . 

 n ^ 



The mean value of r] for the group is then 

 at t' ^e-^'t]', 



at^'Mtis^(^X + ^''V' + ^''V' + ^^-) ''• • • ^^^ 



But by the algebraic theorem of art. 1 in its second form, 

 or Theorem IX. of Chapter XI., since 7i^''' = e''" + ^''i'' + &c., 

 the first of the expressions (5) is less than the second, or 

 the mean value of t] for the group is less at t* when the n 

 systems are all in the same element of extension in phase 

 DV'', than at t" when they are scattered over difi'erent ele- 

 ments. If t' is a later instant than t" ^ we have for this 

 group diminishing mean entropy. If t" is later than t' we 

 have increasing mean entropy. 



17. But no reason is giA-en, and no reason is apparent, 

 why a set of systems, which were in different extensions in 

 phase at an earlier instant, t" , should be in the same exten- 

 sion in phase at a later instant t' , rather than vice versa. All 

 that is proved is that the mean entropy of a group will in 

 certain cases either increase or diminish. To determine 

 whether it tends in general to increase or to diminish, or to 

 prove that it tends to diminish, we require, as appears to me, 



