302-305] Viscosity when the Molecules are Elastic Spheres 249 



we find 



f 00 4 (km)* c 5 e~ hmc * , 



\c = I -j= dc 



J o TTj/o- 2 i/r (c vhm) 



* f 



o- 2 ^o 



and if I is denned by equation (568), we have 



Xc = Ic 

 so that in formula (573), we must take 



/: 



' a? e~ x *dx 



The integral has to be evaluated by quadrature. Tables for its evaluation 

 are given by Tait*. The integral has also been evaluated by Boltzmannf, 

 whose result agrees to three significant figures with that obtained by Tait. 



Using this value for the integral, we find that 



1= 1-051 -f=^ - .. ...... (575). 



The value of I, calculated accurately for our present purpose, accordingly 

 differs by about 5 per cent, from the mean free path, 



1 



V27TI/0- 2 



calculated in 30. 



305. We turn now to the error which has been introduced by ignoring 

 the persistence of velocities. We have found that this persistence is measured, 

 in a gas of which the molecules are elastic spheres, by a numerical factor 

 which is always intermediate between and ^, and of which the mean value, 

 averaged over all collisions, is '406. 



If, on the average, each particle has described a path of which the 

 projection on the axis of z is f, with a velocity of which the component 

 parallel to the direction of the axis of z is w, then, on tracing back the 

 motion, we know that as regards the previous path of each molecule the 

 expectation of average velocity parallel to the axis of z is 0w, where 9 

 measures the persistence, and therefore, on the average, the expectation of 

 the projection of this path on the axis of z may be taken to be 8%. Similarly,. 

 the expectation of the projection of each of the paths previous to these may 

 be taken to be d 2 ^, and so on. 



* Collected Works, n. pp. 152 .-md 178. 

 t Wiener Sitzungsber. LXXXIV. p. 45 (1881). 



