Boltzmaruis Law of Probability e—^'X. 713 



in the same ratio. Then r becomes r+- r r, or ^7^ = r—~; 



dv ov 



and if #, y, z are the rectangular components of r, d#=# ^-, 



with similar values for "dy and B~. Multiply Clausius' equa- 



tion by 2/1,7-. That gives 



hpdv= - 2 7T + '* 2S R 7.- r 

 1 ou 2 h ov 



— ji'd loo- v + A £2 »*^r 



= r t -d\ogv-h d X^ v (A) 



dv 



where -^ denotes the mean value of that function. 

 dv 



Next let the system receive a small accession oF heat, dQ. 



That will be spent (1) in work done in increasing v against 



the external pressure /?, that is pftr ; (2) in increase of the 



mean kinetic energy £, or -~ 7 ~, that is — KTaB^j (3) in increase 

 ° - ' zk 111" o 



of mean potential ^. That leads to dQ=y>c)r — ,,.,<$/' -fd%, 



2/i ~ /r 



and multiplying by -= , 



= |nB log r- 3*^B»- x + |^(*X>- Jx3«, by (A). 



Of these five terms, ?v<3 log r, , , and = ^ (7*^) are complete 



li o 



differentials. In order therefore that may be a complete 



differential, it is sufficient and necessary that the sum of the 



i 2 ~ , 2 7 <;/ v 



two remaining terms, namely -5P l ~o A j 0") he a 



complete differential of some function, w, of h and r. That is, 



2 

 as we may omit the factor^ , 

 u o 



hi -. , c/'/ 



* r 



*W-*2*-2w + £a 



