126 BELL SYSTEM TECHNICAL JOURNAL 



tropy". It remains to interpret the rest of the right-hand member of (19), 

 which I will call "the contribution of temperature to entropy". To do this 

 we must re-enter the momentum-space. 



From (15) and (5) and (8) we get, for the entropy Sm of the flock of dots 

 in the momentum-space: 



Sm = -k\n W = - kN Zwi \n Wi 



= -kNA 2 (In^ - BEi)e~'"'' (21) 



Refreshing our memory from (10), we see that the first term of this expres- 

 sion reduces to —kNlnA. Refreshing our memory from (11), we see that 

 the second term reduces to -f kNBU or kBU. Referring now to one 

 gramme-molecule of gas, I put R for i\'^, and find : 



Sm^ -R\nA -\- kBU (22) 



Sm is hereby given as a function of U, but a more complicated function than 

 appears on the surface, since A depends upon B (equation 10) and B upon V 

 (equation 11). Yet when we differentiate Sm with respect to U, and in so 

 doing take account of these complications, it turns out that we might as well 

 have been oblivious of them! for the result is the same as though A and B 

 were constants: 



dSm/dU = kB (23) 



Now the temperature, which has so often slipped into this argument in 

 ways more or less surreptitious, is about to make its formal and ceremonious 

 entry into the statistical picture. We turn back to equation (17), and 

 deduce: 



dS/dU = 1/T (24) 



The derivative here standing on the left is the derivative of entropy with 

 respect to energ>' under the condition of constant volume: a thermodynami- 

 cist would write it {dS/dU)v ■ It is therefore properly to be identified with 

 the derivative in (23), and we make the two identical by putting: 



B = 1/kT (25) 



Now taking the entropy S to be the sum of Sc and 5„, , we find : 



S = Sc-\- Sm= - RlnA -\- U/T + R]nV - R In Vo (26) 



and this is to be compared with (19), the thermodynamic expression for 

 entropy, which I repeat to make the comparison easier: 



S = I (C^/T)dT + i? In F + constant (27) 



