54 THE PHYSICAL SIGNIFICANCE OF ENTROPY 



" The entropy of a physical system in a particular state depends 

 solely on the probablity of this state." 



No rigorous proof is here attempted, nor any numerical com- 

 putations; for present purposes it will suffice to fix in a general 

 way the kind of dependence of entropy on probability. 



Let S designate the entropy and W the probability of a 

 physical system in a particular state, then the above theorem 

 enunciates that 



S=f(W), ........ (8) 



where f(W) signifies a universal function of the argument W. 

 Now, however W may be defined we can certainly infer from the 

 Calculus of Probabilities that the probability of a system, com- 

 posed of two entirely independent systems, is equal to the product 

 of the separate probabilities of the individual systems. For 

 example, if we take for the first systm any terrestrial body what- 

 ever and for the second system any hollow space on Sinus, which 

 is traversed by radiations, then the probability W, that simultane- 

 ously the terrestrial body will be in a particular state i and said 

 radiation in a particular state 2, will be given by 



(9) 



where Wi, W 2 respectively represent the separate probabilities 

 of said two states. Now let Si, S 2 respectively represent the 

 entropies of the separate systems corresponding to said states 

 i and 2, then according to Eq. 8, we have 



But, according to the Second Law of Thermodynamics, the total 

 entropy of two independent systems is S = Si+S 2 , and conse- 

 quently according to (8) and (9), 



+f(W 2 ). 



