422 



MATHEMATICAL APPENDICES 



32* 



The following series of figures show that the two expressions 

 agree remarkably well both for small and large values of w, while 

 for middle values of u> a regular deviation occurs. 



The values obtained from the approximate formula are rather too 

 large, but the errors are in all cases less than 2J per cent. The 

 simple approximate formula can therefore very well be used instead 

 of the more complicated exact formula in all calculations when 

 absolute accuracy is not desired. 



33*. Molecular Free Path 



With each collision a molecule starts on a fresh rectilineal piece 

 of its zigzag path. The number of collisions is therefore the 

 same as that of the straight bits of the path traversed. Conse- 

 quently we find the mean length of one of these bits by dividing 

 the length of path traversed per unit time (which is measured by 

 the velocity) by the number of collisions experienced per unit time. 



Since mean values are taken in this calculation, our first 

 thought is to divide the mean speed by the mean collision- 

 frequency, and call the quotient the mean free path. In this way 

 we have already obtained in 28* the value 



T' <?- 3 



= A WJV 



for the mean free path of the particles of a simple gas, from the 

 assumption, which is approximately true, that the speeds of all the 

 particles are the same. Instead of this value, which Clausius 1 

 gave, Maxwell 2 obtained the somewhat smaller value 



T =-= 1 



"r V 



1 Phil. Mag. 4, xix. p. 434, 1860 ; Abhandl. U. d. mech. Warmetheorie, 

 Abth. 2, note to p. 265 ; Mech. Wdrmetheorie, iii. p. 61, 1889-91. 



2 Phil Mag. 4, xix. p. 28, 1860 ; Scientific Papers, Cambridge 1890, i. p. 387. 



