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XXXII. On the Statistical Theory of Radiation. 

 By Prof. Sir J. Larmok, F.R.S* 



IN the Philosophical Magazine for July (p. 122) Prof. 

 H. A. Wilson, in a valuable review of my recent paper 

 on the statistical theory of natural radiation!, concludes that 

 its procedure does not- really evade the main difficulty, that 

 an atomic constitution of energy must be implied in such 

 investigations. One of the positions advanced in the paper 

 was that the magnitude of the element of energy needed lor 

 the statistics might be chosen at will, provided the size of the 

 elementary cell was chosen in a fixed proportion to it. 

 Though such a theory has, and must have on the most 

 favourable view, imperfect and provisional features, it does 

 not appear to me that Prof. Wilson has established this 

 formidable addition to their number, and for the following- 

 reason. 



Using his notation, the heads of the argument there set 

 out, perhaps too briefly}', were as follows. If S is entropy 

 and W is the number of ways in which the system can be 

 arranged in the actual state, then 



S=MogW 

 =k\ogW 1 W 2 ...W n , 



where W l5 W 2 . . . are the numbers for the parts of the system. 

 If the first part contains ?? 2 elements of energy each of amount 

 6 b contained in N x cells, and similarly for the others, then 

 the total energy is 



E = e^! -f e 2 n 2 + ... € r n r . 



The natural state of an isolated system is the one that makes 

 S maximum subject to E remaining constant. This requires 



fi u^i ~ e 2 ~dn 2 

 where 6 1 Sn 1 = SE 1 , .... Thus 3 is a quantity the same for 

 all the parts of a system which is in equilibrium of exchanges 

 of energy : in fact if absolute temperature T is defined by 

 the Clausius formula dS = dE/T, then 3 is T" 1 . Also the 

 working out of the actual value of Wi leads to 



* Communicated by the Author. 

 t Roy. Soc. Proc. 1909, vol. lxxxiii. A. pp. 82-95. 

 X Errata should be noted : on p. 92 $ should be T ; on p . 93 the factor 

 8tt/A 4 should be A 4 /8t. 



