Systems, and Planck's Theory of Radiation. 947 



A, B, C ... are independent events. We put S = & log W, 

 and S is then Boltzmann's measure o£ the entropy, proba- 

 bilities now being measured on the basis provided by the 

 generalized space. 



6. Let E„ E 2 , ... be the energies of those parts of the 

 system with which the properties A, B, C... are associated, 

 and let E be the total energy given by 



E=E 1 +E 2 + (3) 



The total entropy S is given by 



S = HogW A + HogW B + . . . -I HogK. ... (4) 



The characteristics A, B, 0... may be chosen so as to 

 determine the partition of energy. To be precise, let 

 characteristic A be satisfied if E x lies between E^ — -^ and 

 E/ + -J6! ; let B be satisfied if E 2 lies between E 2 ' — ^e 2 and 

 E 2 / +2 6 25 an d so on. Let it be assumed, as a property 

 of the system, that if left to itself, it will assume a state 

 in which the energy is divided in a definite manner, namely 

 one in which E 1} E 2 , ... become equal to E/, LV---, 

 at least to within small ranges e 1? e 2 , ... Then W must be 

 equal to unity for these values of E 1? E 2 ..., and this is not 

 only the maximum value of W, but is greater than the sum 

 of all other values. It follows that S also must be a maximum, 

 when Ej, E 2 , ... have the values E/, E 2 ', ... subject to 

 condition (3). The analytical condition for this is, in the 

 usual way, that E/, Eg', ... shall be given by the system of 

 equations 



BS___BS _ 



3Ei~BE.~ w 



combined with equation (3). 



We can find the value of each fraction by supposing that 

 part of the system is a perfect gas. We may assume this 

 part of the system to obey the Newtonian laws, so that 

 its co-ordinates ~P 1} P 2 , . . . P m may be supposed identical 

 with its Lagrangian co-ordinates and momenta, and its 

 energy E x will be of the form 



Bi=S«iPA (C) 



the sum extending to m terms. The value of W A is now 

 proportional to the volume of the region of the generalized 

 space in which X«iPi 2 lies between Ej — ^e x and Ex + ^e^ 

 and is therefore of the form cW m ~ 1 e 1 , where c is a constant. 



