for the Theoretical Proof of Avogadro' s Law. 317 



In this /=/(», 0, fi=f(V,t), f=f(v',t), .A'=AV',0, 

 F=F(V), F=F(«',t), F 1= F(V,0, F^F^', i). A is 



the diameter of a molecule of the second kind. 



By means of these equations we can show that the quantity 



E =j /(//-iy^+f FifZFi-lJVW . . (21) 



*^o Jo 



(whose intimate connection with the magnitude called by 

 Clausius "Entropy" I have remarked elsewhere) can only de- 

 crease or (in an extreme case) remain constant. I denotes 

 the natural logarithm. We have 



f-J>lf-* + J>? ,Wj 



dE 



* 



therefore, with reference to the equations (20), 



J_BE 

 2?r 'dt 



IT 



(too ^,oo „ n „2 /»2ff 

 I 1 I 1 Wf.(ffl-ffi)v*V*rTsadvdV<nid&dO 

 ~ o Jo Jo "o Jo 



+ I I III AHF^WFJ-FFiyYZrTsadvdYdTdSdO y (22) 



^ o Jo Jo Jo */o 



IT 



+ C C^ 2 ^^nfif'F^-fF^Y^TsadvdYdTdSdO 



Jo Jo «^0 «>0 ^0 



IT 



(ico /»oo /^jr ^,o f*2ir 

 I I I I 8*lF 1 (fF l '-fF 1 yv*rT8*dvdV<n!d&dO 

 i Jo Jo Jo Jo 



From the circumstance that every impact may take place 

 also in inverted order, it follows at once that the value of anv 

 definite integral which, according to the above, is to be 

 extended over all possible values of the variables before 

 impact, does not alter if we exchange the value of the variables 

 before impact for their values after impact, and vice versa, 

 and, finally, again integrate for all possible values. 



We shall have, therefore, for each function \P compounded 

 in any way of the variables under the functional sign, 



