for the Theoretical Proof of Avogadro's Law. 319 



line of equation (22) ; and we have now to substitute the 

 arithmetic mean of all four values. If we treat in the same 

 way the second line of this equation, we obtain finally 



2_dft = rC x C"C2 f %«y 9rTM . dv d y d T dS dO 



v dt J J J Jo Jo 



For the condition of equilibrium of energy, /and F and 

 also E must therefore be independent of the time ; we must 



*7F 

 therefore have -t-=0. But we see at once that the last-found 

 dt 



integral for -7- represents a sum of infinitely small members, 



which are all negative or at most equal to zero ; for if 



ff 

 f'fi—ffi is positive the factor ljj-j, is negative, and vice 



versa. 



dW 



Therefore — can only vanish when each of these members 



itself vanishes. 



If the molecules of the first kind were very small in 

 comparison with those of the second kind, then would 

 A,=0, A = 8. We may assume, still more generally, that 

 the molecules of the first kind are perfectly permeable for 

 each other, and so also those of the second kind for each 

 other, and that each of the latter molecules onlv is surrounded 

 by a sphere of radius 8, at which the centres of the molecules 

 of the first kind are reflected like infinitely small elastic 

 spheres; in the latter case we should have A=A,=0. As 

 soon as 5 differs from zero, f F/ must always be equal to 

 /F, for all values of the variables under the functional sign. 

 Since v, V, and v 1 are quite independent, and only V' is 

 determined by the equation of energy, we find, without 

 difficulty, 



f=Ae- hmv \ F = Be- kMY \ 



But, in consequence of the impact of molecules of the first 

 kind with those of the second, there will therefore be pro- 

 duced Maxwell's distribution of velocities and equality of mean 

 energy amongst all the molecules. I take the present oppor- 

 tunity of remarking that I do not understand how gravitation 



