Law of Distribution of Energy. 151 



redistribution of S among them in the following way. Con- 

 sider any group of N systems for which the variables are 

 #11, #2i? &c, and for which S has the values S b S 2 . . . S N . 



S S 

 Let these contribute respectively ^, ~, &c. ; and let the 



contribution t^ be represented by the energy of a similar 



system whose variables are — ~, ~7^, &c. ; and let —■ &c. 



be represented in the same way. 



From that group of N systems we will form the variables 

 $i . . . a? n ' for one system in the new distribution by making 



, _ #11 +#21+ . ■ . + # Nl 

 #12 + #22 + • • • + %« 



* 2 = " ~7W ""' 



&c. &c. 



see post. (11). 



Now let us take at random other groups, each of N systems, 

 and deal with them in the same way. And so on until every 

 original system has appeared N times as a member of a group 



of N. Since in case of each group it loses ^ , it will finally 



have parted with all its energy S. When this process has 

 been carried out, the whole of S for all the systems will have 

 been redistributed among them. And owing to the mode of 

 formation of x-l &c, the chance of the variables a? x ' . . . a?J of 

 any system in the new distribution having the values 



_nS 



Ci . . . Ci + dcx &c. is Ce 2T dc^ . . . dc n , and the aggregate energy 

 will not have been altered. 



(9) If, then, for the original systems 



/(a?i . . .0=Ce"2T, 



the law of distribution is the same for the new as for the 

 original system. In other words, it is unaffected by our 

 process of redistribution. 



If for the original systems /(#i . . . #J have any other 

 form, we shall choose a ly & 12 , &c, to satisfy the equations 



_ 2 D n D 12 



Xx* =~lJ > #1#2 = -jy > & ' 



