1892.] in a system of moving molecules. 325 



approximately, the volume is 



7-0 x 10 3 -9^1 dPd V d£ u dS 



= e « 2 . ' . udt . -. — cos 6 



,,1 a 3 4>7r 



cubic millimetres. Now a cubic millimetre of hydrogen at atmo- 

 spheric pressure contains about 976 x 10 16 molecules. Hence the 

 number of molecules in do 2 is 



. n ,-„. -9H dPd-ndt 1± dS n 

 6-8 x 10 20 .e a« . , * . udt . -r— cos 0. 

 a 3 47r 



Suppose u to be equal to k times a which is in hydrogen equal to 

 7"62 x 10 5 millimetres per second; we get the whole number of 

 molecules in do 2 to be 



K _ -911 dPdr/dt ,-„„, .dS n 



5-2 . e a» . , . W 6 hdt -j- . cos 0, 

 a 47T 



tZi being measured in seconds ; and this number must be large 

 in order that equation (2) may accurately give the number of col- 

 lisions of the given kind. 



The smallest value which we can ascribe to dt will depend 

 upon the magnitude of the limits d%, drj, d£, dS which define the 

 encounter. Suppose that d% — drj = dt, = a/1000 ; suppose also that 

 dS/4<Tr = 10" 3 . Let us also suppose OP not to be greater than 

 2a, a supposition which excludes less than 0'5 per cent, of the 

 whole number of molecules. Suppose also that k is greater than 

 10~ 2 , so that the relative velocity u is not less than a/100. These 

 suppositions give the whole number of molecules in do 2 to be 

 greater than 9"6 dt . 10 10 , so that this number is more than a 

 million if dt is not less than 10~ 5 of a second. This estimate of 

 the limiting value of dt is perhaps too small as we have taken 

 the limits of d^drjdt, exceedingly small, but it will appear that 

 dt must not be taken indefinitely small and should at any rate 

 be greater than the mean time between collisions, which is of the 

 order 10~ 9 of a second. With the above proviso as to the value 

 of dt, it appears that result (2) can be taken to be accurate, and 

 since result (3) is merely an application of result (2) to the re- 

 versed medium it appears that the assumptions made can be 

 relied on. 



2. It has throughout been assumed that the distribution of 

 velocities is " steady ", and the proof shows that Maxwell's law 

 gives the only possible steady distribution. It is however desirable 

 to show if possible that the system must ultimately acquire a 

 steady distribution. Now the steadiness of the distribution has 

 been assumed twice, first when the assertion is made that the 



