31* MOLECULAR FREE PATHS 419 



where a } and a 2 denote the speeds of flow of thp two groups in 

 the direction of u. 



By substitution as before in the expression for y of 



Ui = U + ill n 2 = U ill 

 0, = S3 + 1*> v a = SB - i 



w l = 353 + iw t^ 2 = SB in> 

 a t = v4 + i& a<2 = A i, 



y assumes its former shape, while r and become 



r= V(u 2 + tt 2 + tt> 2 ) 

 AY -f S3 2 + SB 2 } + Pm{(u - a) 2 + tt 2 + n> 2 } . 



If we now introduce polar coordinates, the integrations with 

 respect to U, S3, 2B are easily performed, and those with respect 

 to u, t>, w partially so, the final shape of the integral being 



o 



which by development in powers of a gives 



7 = 2 



and this for a = reduces to the known result 



7 = s/2a 



If we also develop the exponential function in powers of a we 

 obtain 



7 = 



which shows that the collision-frequency is increased by the flow 

 of the gas by an amount which is of the order of the square of the 

 difference a = a } a a . This difference, or the relative velocity 

 of two neighbouring layers, is in the theory of internal friction 

 always looked upon as very small, and its square as therefore 

 negligible. Here, too, it is a very small magnitude of the order 

 of the molecular free path ; and in the formula, which by 

 introduction of the mean speed becomes 



we may neglect the correctional term as vanishingly small, and 

 therefore apply to a flowing gas the same formulae for the 



E E 2 



