for the Theoretical Proof of Avogadro's Law. 313 



Since no direction in space, and also no element of volume,, 

 has any advantage over another, we may assume that the dis- 

 tribution of energy also remains uniform for all succeeding 

 time. But in general this is prevented by the collisions- 

 which occur. At the time t, let there be in the unit volume 

 4:7rv 2 f(v, t) dv molecules of the first kind of gas whose velocities 

 lie between the limits (6), F(t>, £) having a similar signifi- 

 cation. 



Evidently the problem is conceived in its utmost generality 

 if we imagine f(v,o) and F(V, 6) having any given values, 

 and determine the changes of these functions in course of 

 time. Evidently we have first of all to determine the increase 



,MM) and e **m 



3* ^t 



which the functions/ and F undergo, whilst the time increases 

 from t to t + 6. During the element of time 6, let n molecules 

 in the unit volume out of the 47ry 2 /(v, t) dv molecules of the 

 first kind therein contained, whose energies lie between the 

 limits (6), enter into collision with other molecules of the first 

 kind, and N molecules with those of the second kind. Let 

 us imagine 6 so chosen that n and N, although large numbers, 

 are yet small in comparison with 4t7rv 2 f(v, t) dv. 



Since the number of those molecules for which the velocities 

 after impact also lie between the limits (6), or for which the 

 velocity of the second impinging molecule lies between the 

 same limits, is of an order higher only by an infinitely small 

 amount, we may assume that the velocities of all these n mo- 

 lecules and also of the N molecules after lapse of the time 6 r 

 no longer lie between the limits (6). w + N is therefore the 

 number of those molecules whose velocities were at the be- 

 ginning of the time 6 between those limits, but at the end of 

 that time were not between the same limits. 



But during the time 6, other molecules whose velocities 

 were not previously between the limits (6), in consequence of 

 impacts acquire a velocity lying between these limits. Let, 

 then,p molecules of the first kind acquire a velocity lying 

 between these limits by impacts with other molecules of the 

 first kind, and P molecules of the first kind by impacts with 

 molecules of the second kind. Then 



47n> 2 0^^=j> + P-w-N. ... (8) 



We found before the expression (3) for the number of 

 impacts which in unit time and unit volume so occur that the 

 variables v, V, T, S, and lie between the limits (2). 



