FOUNDATIONS OF THE KINETIC THEORY OF GASES. 73 



The value of the factor in brackets is easily seen to be 



_dj l&V (to 1\ 



dh "*" 3v 2 dh 2 ^\3h^~ 2h 2 v) £ ' 



where v= \/ t~ hv *dv, 



and thus it may readily be tabulated by the help of tables of the error-function. 

 When v is very large, the ultimate value of the expression is 



4 v ; 



77 

 J? 



which shows that, in this case, the " special " state of the particles in the layer 

 does not affect its permeability. 

 10. Write, for a moment, 



— eSx 



as the logarithm of the fraction of the particles with speed v which traverse the 

 layer unchecked, Then it is clear that 



—ex 

 £ 



represents the fraction of the whole which penetrate unchecked to a distance 

 x into a group in the " special " state. Hence the mean distance to which 

 particles with speed v can penetrate without collision is 



/ £ xdx i 



e~ ex dx e 



/n 



This is, of course, a function of v ; and the remarks above show that it increases 

 continuously with v to the maximum value (when v is infinite) 



i.e., the mean path for a particle moving with infinite speed is the same as if 

 the particles of the medium traversed had been at rest. 



11. Hence, to find the Mean Free Path among a set of spheres all of which 

 are in the special state, the natural course would appear to be to multiply the 



VOL. XXXIII. PART I. K 



