324 Prof. L. Boltzmann on the Assumptions necessary 



R, in each of which let there be an equal number of molecules 

 A constituted as before, and each with the same distribution 

 of velocities. Let A7rv 2 f(v)dv be the number of molecules in 

 the unit volume whose velocities lie between the limits (6) . 



In each of the vessels R let there be a single molecule B 

 which in each vessel shall be now here, now there, now moving 

 in this direction, now in that with equal probability. Let the 

 number of vessels R for which the velocity of the molecule 

 B shall lie between the limits (7) be 47rNV 2 RF(V)dV, where 

 N is the number of all the vessels, of which each has the volume 

 R. Further, let there be no other impacts than those of the 

 molecule B in each vessel with the molecules A. Then evi- 



dF . 



dently, at least for the first moment of time, ^— is determined 



by the same equation by which ^ ? ' was determined 



above, since it is quite indifferent whether all the molecules 

 B are in the same vessel or whether each is in a separate one. 



But if we now imagine all the N vessels of volume R brought 

 together into one, we obtain a vessel of volume NR in which 

 in the unit volume there are 47rV 2 F(V)<iV molecules of the 

 second kind whose velocities lie between the limits (7) . In 

 order that this may not be a proper fraction, but a very large 

 number, we may imagine the unit volume to be very small in 

 comparison with R; that is, R as any large number, which 

 must, however, be still very small in comparison with N. 



The change in /will no doubt be different in the different 

 vessels ; let us, however, denote by /the arithmetic mean of all 

 the values of/ for the different vessels; then again, at least in the 



first moment of time, J- wn "l De given by the same equation 



as before, 4' . In the former expressions we have of 



course A, = A = 0, since neither the molecules A impinge upon 

 each other nor the molecules B. We convince ourselves most 

 easily of the truth of the above assertion by imagining all the 

 vessels R united into one large vessel of volume NR, in which 

 in the unit volume there are 47rV 2 F(V)<iV molecules B whose 

 velocities lie between the limits (7), and 4:7rf(v)v 2 dv molecules A 

 whose velocities lie within the limits (6). We may now sus- 

 pect that in course of time / assumes different values in dif- 

 ferent vessels, and not until the end becomes again equal in all. 

 This suspicion is most easilv removed by supposing that 

 at the beginning of the time / and F have the values denoted 

 above by / and F^ Everything then remains as it was ; 



