for the Theoretical Proof ofAvogadro's Law. 321 



x and X, ty and W. In place of x' we have X', the energy of 

 the second molecule after impact. 



We have further, except for a constant factor, 



E= 1 \/x$(x, t)[l(f>(x, t) — l]dx 



+ 1 \/X®(X,t)[l®(X,t)-l]dx, y (24) 



Jo 



§ =£va*(*, 0%^-+ J o VS*(X, 0^=|% 



Before we substitute the above values in equation (24) } 

 we have still to establish two properties of the functions -^, 

 M/ 1 , and %. In the functions yjr and ^ both impinging mole- 

 cules play the same part. It is then just as probable that 

 before impact the energy of the first should lie between x 

 and x + dx, that of the second beween X and X + dX, and, 

 after impact that of the first between x' and x' + dx', as that 

 inversely before impact the energy of the first molecule should 

 lie between X and X -f dX, that of the second molecule between 

 x and x + dx, and after impact that of the second between x' 

 and x' + dx'. Therefore that of the first lies between x + X— x' 

 and x + X — x' + dx'. Or, in algebraic language, 



+(*, X, x') =f(X, x, x + X-x ! ), . . . (25) 



V(X,x, X f )=V(x, X, x + X-X 1 ). . . (26) 



The second property may be obtained as follows. We found 

 the value (3) for the number dZ of impacts which occur in 

 unit time and volume between a molecule of the first and one 

 of the second kind in such a way that the variables v y Y, T, 

 S, lie between the limits (2). We will first introduce the 

 variables r, G instead of Y and T. Then, instead of G, twice 

 the energy of both molecules, 



y = (m + M> 2 + Mr 2 -|- 2Mvrg = (m + M)t/ 2 + Mr 2 + 2Mv 'rg'; 

 this gives 



47T 2 



dZ= -^vr 2 sadvdrdydSdOf{v)F(Y)8 2 . 



We will now for the constants v, r, y, and S introduce the 

 variable xf instead of 0. Since in this g is also constant, 



2 , 2 , 4M^ , 2 , , 4MW 



™ + M27rVrS 2 

 70) 

 Phil Mag. S. 5. Yol. 23. No. 143. April 1887. 



aVL = ^P *\_ dvdv ! drdyd$f{v) F (Y) . 



