Theory of Radiation. 45 



a continuous though not necessarily an unrestricted range 

 of variation. If discontinuous changes are admitted we 

 must suppose them to involve a transformation in the nature 

 of these variables. The values of the new and the old must 

 be connected by equations sufficient to determine the one 

 when the others are given. 



I write dN for the probability of a material state in which 

 X1X2 ••• ®n have values lying within the infinitesimal intervals 

 dx\, dx 2 . . . dx a . 



d$=p'dV.. . .| 



dV = doc l dx 2 . . . dx n - I 



Let a material system change from the time t 1 to the time 

 t 2 , whether continuously or not. 



(tfN) 2 =(<7N) 1; (4) 



where the suffixes denote the values of dN at t 2 and at t^ 

 Then the most general solution of (8) and (4) is given by (5) 



p'=e-**xp, (5) 



where p is any particular solution and h is an arbitrary 

 constant. The statistical method leads to (5) even for laws- 

 of change so general as those here assumed. H is the total 

 energy, and it is taken for granted that the only quantities 

 invariant throughout all change are functions of H. 

 Since p' and p are both solutions of (3) and (4), 



p'/p is therefore an invariant, and 



p'= P xf(R) (6) 



The essential postulate of the statistical method is that for 

 the purposes of the theory of heat any finite body can be 

 divided into finite physically independent parts. Call these 

 parts A and B, 



By (6) dN A =f{Il A )p A dV Ay dN B =f(H B )p B dV B , 



d$=f(B A )f(IT B )p A p B dV A dY B . . . . (7) 



Now (/Oa^V a ) 2 = (p A dY A )u 



(pBdYB)-2=(pJidYB)l' 



Hence (pAPBciY A dY B ) 2 =(pApBdY A dY B ) 1 . 



