656 Mr- G. H. Livens on the 



allotted to the element 8\*; if, however, the point be assumed 

 to be represented by a small portion of the line o£ length e, 



then its chance of receiving the energy is kt; in that case, 



however, there is only a finite number of such points in the 

 length S\, viz. 



N - — 



so that the chance for the particular wave-length in the 

 simpler system represented by the small length e varies in- 

 versely as the number Nx of such, which it is necessary to 

 superpose to secure absolute continuity in the complete 

 spectrum. But according to the above this chance is measured 

 by a\, so that we see that 



We shall find that this result is necessary for the final 

 determination of our formulae t« 



4. The general theorem of partition of energy. 



The problem is now to determine the new form of the 

 general theorem of partition of energy which will make it 

 applicable in cases of the type of those under discussion. 

 We must determine the most probable distribution of the 

 energy among all the various types of coordinates, each with 

 its own definite chance of receiving a specified amount of 

 the energy. This problem is most easily solved in the method 

 of Planck, as a problem in molecular statistics by dividing 

 the energy into small differential elements e (to be made 

 ultimately zero) and then determining by direct counting 

 the most probable distribution of these elements in the 

 various coordinates, divided, say, into (r + 1) sets of specified 

 types containing numbers n Y n^ n 2 ... n r — with respective 

 extents 1, a l5 « 2 , ...a,. We have to find p, p x , p 2 , ,.. p r , 

 which are the numbers of the elements of energy that, in the 

 ultimate state of statistical equilibrium, will reside in these 

 groups of similar coordinates respectively subject to the 

 conservation of its total amount E. The chance of the dis- 

 tribution just expressed being made up of independent 



* "When it acts as a representative point in the simpler system, 

 f It is in fact the condition implied in the existence of a definite 

 formula which requires that some relation of this sort must hold. 



