672 



and integrating partially as regards the second term under the 

 integral* sign, (6) changes into 



5 = SNl' 



4 r §»(/§ ^ 



— I In 



— ln(l— ^-•^) 



For large values of a; this gives 



S = ^Nk : - 3,r :S e-'" 



15 .f' „=i 



14 8 8 



_ J 1 \ 



I *> 9 ' r. a ' 4 4 



n.v n'x 11 X n X 



for small values of x 



^ ^ 3 8^52! 74!^ 



(8) 



(y) 



.(10) 



if ^1, B^ . . . . denote the Bernouillian coeflicients. 



«. Loio temperatures. If of the development (9) we write down 



n 



the first term only, we may put '^^ = ^ with 6^ according to Suppl. 

 No. 30a equation (18a) or (186). This gives 



S — aMWT\ (11) 



if M denotes the molecular weight, and « a constant which depends 

 on Planck's constants A and h and on the Avogadro number. 



According to (11), the volume being kept constant, the entropy 

 approaches to for 7'=0. It does so proportionally to T\ which 

 is in agreemeiit with (2). The latter would not have been the case 

 if (see Suppl. No. 'èOa § 4c) the zero-point energy had not been 

 introduced in the theory, cf. H. Tetrode, Physik. ZS. 14 (1913); p. 214. 



^. Hkjii temperatures. Retaining in the development for high tem- 

 peratures only the first term which gives a deviation from theequi- 

 partition laws, we obtain : 



5 = 



2 3 T 1575 T- 



or 



S = m j Y In /? MTV'h + 4 + ^ (/? MTV'ky^- 



(13) 

 (14) 



where /? is a constant depending on //, k, and N- 



3 

 The additive constant Nk{-^ -{- -Ur^M), with which the '^cliemi- 



eal constant" is connected, agrees with the expression obtained for 



it by Tetrode I.e. without the assumption of a zero-point energy. 



b. From (11) and (14) the entropy of the molecular quantity ot 



