450 MATHEMATICAL APPENDICES 48* 



w 3 l for the whole of the N molecules in unit volume is to be 

 found, wherein / denotes the length of a path already begun, and 

 W the speed with which it is traversed. Thus for each particle 

 we are concerned only with the single path which has been begun. 

 In 38* we also formed another mean value by considering 

 for each particle all its paths traversed during a lengthened 

 period. If we also now take into account all the paths traversed 

 in unit time, we have to bring into the calculation B paths instead 

 of a single path. This process results in a larger mean value, 

 since the number B and the free path I are (by 37*) the larger 

 the larger the speed w. This second mean value is denned by the 

 formula 



which shows that our new consideration is simply equivalent to 

 replacing the collision-frequency B by its mean value T. The 

 formula leads to the value 



If we put this in the formula for the coefficient of viscosity, we 

 obtain the equation 



77 



first given by Stefan, 1 which gives too great a value. 



On the other hand, we get too small a value for the coefficient 

 if we substitute for the variable free path Z its mean value L, that 

 is a length which is too great for small values of w, and too small 

 for large values of w, so that in the calculation of the mean the 

 smaller values come more into account than the larger. We have 

 therefore 



or 



77 > 



For the value of the coefficient K-, therefore, we obtain the 

 following limits : 



<<* 



or 0-27 <K< 0-39, 



1 Wiener Siteungsber. 1872, Ixv. Abtb. 2, p. 363. 



