﻿the Partition of Energy. 829 



which are quite independent, then by a fundamental principle 

 of probability, the joint probability is the product of the 

 separate probabilities ; that is, 



WxW.-Wa (4-1) 



On the other hand, the joint entropy is the sum of the 

 separate entropies, and so 



S 1 + S 2 =S ]2 . ...... (4-2) 



Then to satisfy the functional relationship we must have 



S=*logW, (4-3) 



k being a universal constant. Next, to evaluate W, a 

 definition is made of " thermodynamic probability " as the 

 number of complexions corresponding to the specified state : 

 this is made a maximum subject to the condition of 

 constant energy, and the maximum of k log W is equated to 

 the entropy S, which is then shown by examples to be the 

 entropy of thermodynamics. Observe that there are two 

 separate processes involved : in the first the determination 

 of the maximum fixes the most probable state of the 

 assembly by itself. In the second the assembly is related 

 to the outside world by determining its entropy ; and then 

 the absolute temperature scale is introduced by the relation 

 dS/dE = l/T. 



Now there is much to be criticized in this argument. In 

 the first place there is a good deal of vagueness as to what 

 is happening. For the addition of entropies can only be 

 realized by some form of thermal contact *, and is then 

 only in general true when the temperatures are equal ; and 

 both these conditions require that the assemblies shall not 

 be independent. So it is only possible to give a meaning 

 to (4*2) by making (4*1) invalid. Again, without more 

 definition the probability of a state is quite ambiguous: 

 for example, we can speak of the probability of one par- 

 ticular system having, say, some definite amount of energy, 

 and for independent assemblies (4*1) will be true of this 

 type of probability, but it will have no relation whatever 

 to entropy. This objection is supposed to be met by the 

 definition of "thermodynamic probability' 5 ; but that is a 

 large integer and not a fraction, as are all true probabilities, 

 and so (4'1) cannot be maintained simply as a theorem in 

 probability. 



* There is perhaps an exceptional case, that of radiation, worked out 

 by Laue and cited by Planck (' Radiation Theory/ ed. 3, p. 116) ; but a 

 general theorem must be generally true. 



