1085 



rni-2 r 

 G =:= —;:r; I dx^ dx^ d,i\ dx. dx. dx 

 'J 





£<v/i 



when we- introduce x \/in = a\, y \/m = ,t\, z \/m = x^, x V'^f= x^, 

 vVf— '^5 and zVf = x„ so that f = x,^ + x^ + x;- + x: + x,^ + x^\ 

 The integral occurring in G, therefore, represents the content of 

 a sixdimensional sphere with a raduis l/i'A, and is therefore pro- 

 portional to (r^)^ Bearing- in mind that v = — / '—, we see that 



2jr y m 



G and with it also H, becomes an absolute constant. 



If we assume for a linear vibrator that besides \ibrationy with 



a frequency v rotations occur with a frequency v' = , it 



appears here in the same way that G and H become absolute 

 constants. 



Hence we see that on these simple suppositions Planck's supposition 

 about the zero point energy directly leads to Nernst's heat theorem. 



As known Planck formulated Nernst's theorem by assuming that 

 the entropy remains finite at T = 0, and does not become — go, as 

 it would have to do according to the older theory. According to 

 the older theory, e. g. accordii>g to Boltzmann, one would have to 

 come to the value -- gd, because at 7^= the molecules would all 

 have a velocity zero, and tliere would, therefore, be only one 

 possible distribution of the points of velocity in the diagram of 

 velocity. At every higher temperature there* would be oo many 

 velocities possible for every molecule ; there would therefore be 

 infinitely many possible distributions of the points of velocity. The 

 probability at higher temperature would therefore be oc times as 

 great as at T=0, which leads to an oo difference of entropy. 



It is interesting to observe how the two suppositions introduced 

 by Planck into physics evade this difficulty and make the entropy 

 difference finite in the two only ways possible. The infinite entropy 

 difference could namely be evidently evaded in two ways ; namely 

 1. by assuming that there is a finite number of distributions of the 

 points of velocity also at high temperature, and 2. by assuming that 

 there are infinitely many also at T=0. The former hypothesis is 

 that of the energy quanta, the second that of the zero point energy. 

 Each of these two suppositions leads to a finite relation of the number 

 of possible distributions at T = and at 7'^ 0, and hence to a 

 finite entropy difference. 



Let us now examine the distribution of the energy at higher 

 temperature. We shall continue to assume that a number of mole- 



