ON THE KINETIC THEORY OF GASES. 209 



will be interfered with, as regards at least the velocities 

 of translation (see. post 34). Condition (2) is assumed to 

 be always satisfied by the molecules of a very rare gas. 

 As the gas becomes denser, we arrive at a state in 

 which no molecule or group of molecules is ever free from 

 the influence of other molecules not belonging to the 

 group. Condition (2) now fails, and the proof, as above 

 given, fails, whether the law continues to hold or not. 



13. For the above conditions Boltzmann would substi- 

 tute the following, namely, that a single system shall, if left 

 to itself, at some time or other pass through every combina- 

 tion of the co-ordinates and velocities which can be reached 

 from its initial state consistently with the conservation of 

 energy. And he would, I think, express the law by saying 

 that on average the time durino- which the co-ordinates and 

 velocities are between the limits " a " and " b " above varies as 

 € -fc<x+T) dq^ m , dp n . It seems to me to be open to question 

 whether this method would succeed in any case where the 

 other would fail. See an interesting paper of his in the 

 Philosophical Magazine for March, 1893, in which he dis- 

 cusses a test case suggested by Lord Kelvin. 



14. For the present it is sufficient to point out a par- 

 ticular consequence of the i - h(x + r) distribution, namely, 

 that if a eras consists of stable molecules, each havino- n 

 co-ordinates, including three of position in space, then 



enero-y of translation 3 ,_ .„ . . 



— i — r^r~- — — = — on average. (It will be ob- 



whole kinetic energy u & v 



served that we have at this point introduced a fourth con- 

 dition besides (1), (2) and (3), namely, that the molecules 

 are to be stable systems.) To prove this, consider that T, 

 the kinetic energy of a molecule, is a quadratic function of the 

 velocities of the form 2T = m (x 2 + y 2 + z 2 ) + a x q* + b I2 q I q 2 

 + etc., where x, y, z, are the co-ordinates of position, 



• • • • 



q 1 . . . q n _ v the remaining co-ordinates, and x y z q I . . . 

 <7„_ 3 , etc., the corresponding velocities, and a x b I2 , . . . etc., 

 are generally functions of the co-ordinates. 



Now, according to the theorem, if we consider all those 

 molecules for which the co-ordinates x . . . q H . . . etc., 

 have given values, but the velocities may have any values 

 between + co and — 00 , we can find the mean values of 



