444 MATHEMATICAL APPENDICES 45* 



where p and G, as before, denote the density and mean speed of the 

 molecules. If in this formula, which was first deduced by Max- 

 well, 1 we put the value of the mean free path given byClausius' 



theory, we obtain 



77 = m(7/4:7r5 2 . 



These expressions agree with those obtained in an elementary 

 way in 75 and 76. 



46*. Calculation ,of Viscosity on the Assiimption 

 of Maxwell's Law of Distribution of Speeds 



If we put on one side the inaccurate assumption that all the 

 molecules have equal speeds, and replace it by Maxwell's law, 

 as in our calculation of the pressure, we can at first introduce a 

 simplification which is sufficient as an approximation. If the 

 speed with which the gas flows is so small, as it is always assumed 

 in the theory of viscosity, we can consider it as negligible in 

 comparison with the very rapid motion of the molecules. It is 

 then, therefore, allowable to employ Maxwell's law in the form 

 which, strictly speaking, is valid only for the state of rest. In this 

 manner Boltzmann 2 and Tait 3 have calculated the value of 

 the coefficient of viscosity. 



We arrive at the value of the momentum carried this way and 

 that in unit time across the surface element dy dz in the same 

 manner as in the foregoing calculation. In unit volume there 

 are 



particles with speed w, and therefore 



4r~*tf (Jbi) V*""V4w r 2 dr sin s ds d$ 



in the volume-element r 2 dr sin sdsd<f>', of these there pass over in 

 the direction of dy dz, given by the angles s and ^>, the number 



dy dzN(km/Tr)^e~ Hma> *Mio dr cos s sin s ds dy. 

 Each particle begins B new paths in unit time, where 



B S 



1 Phil. Mag. [4] xix. p. 31, 1860 ; Scientif. Works, 1890, i. p. 390. 



2 Wiener Akad. Sitzungsber. 1881, Ixxxiv. Abth. 2, p. 41. 



3 Trans. R.S.E. 1887, xxxiii. p. 259. 



