Probabilities to the Movement of Gas-Molecules. 243 



powers o£ u magnitude which is a function other than linear 

 of numerous independently fluctuating elements. 



These conclusions and those which follow are readily 

 extended from two to three dimensions. 



2. The constants of the frequency-function. — The form of 

 the sought function having been ascertained, we have next 

 to evaluate the constants. For this purpose we might obtain 

 aid from one of the other arguments, from the third as before 

 (/. c. pp. 252, 263), or from the second (below II. 1) if 

 employed to determine the consta?its, the function being- 

 given. Otherwise, supposing the ultimate stable distribution 

 of velocities to have been reached, observe the velocities in 

 any assigned direction, e. g. U, for numerous specimens 

 presented at random by molecules of the type M. Their 

 average velocity in that direction ought to be the same as 

 the average of any other large set of U-velocities, in parti- 

 cular the set which is formed by molecules of the selected 

 class meeting with molecules of the m type. This condition 

 requires certain relations between the coefficients of the 

 frequency-function. In the simplest case where the function 

 is of the Form (1) the U of any M particle colliding with 

 any m particle becomes by (3) L T/ , where 



TJ' = (M.-m)U + 2mu. 



Since [XT' 2 ] = [U 2 ], square brackets denoting mean value, 

 we have 



[U 2 ] 2 =(M-m) 2 [U 2 ]+4m 2 [t6 2 ]; ... (4) 



the mean value [Ui«] being zero (if the mass-centre of the 

 system is zero). Now from (1) we have 



[U 2 ] = 1/2A, and [t*»] = l/2a. 



Substituting in (4) and multiplying by 2A, we have 



l = (M-m) a + 4m 2 A/a (if M + m = l). 



Whence a\m— A/M. Thus the index of (1) is of the form 



-X(MU 2 +mw 2 ) (5) 



If the frequency-function is of the form (2) substitute 

 1/A(1 — r 2 ) and l/a(l — r 2 ) for 1/A and 1/a in the preceding- 

 argument ; and a similar conclusion will be reached. The 

 index will now be ol the form 



-A(MU 2 -2rVM^UM-m M 2 )/(l-r 2 ). . . (6) 



For the general case of molecules with several degrees of 

 freedom we may use the formulae for the components of 

 momentum after impact between such molecules (/. c. p. 261). 



R2 



