FOUNDATIONS OF THE KINETIC THEOKY OF GASES. 71 



II. Mean Free Path among Equal Spheres. 



6. Consider a layer, of thickness Sx, in which quiescent spheres of diameter 

 s are evenly distributed, at the rate of n x per unit volume. If the spheres were 

 opaque, such a layer would allow to pass only the fraction 



l—n 1 irs 2 Sx/4 : 



of light falling perpendicularly on it. But if, instead of light, we have a group 

 of spheres, also of diameter s, falling perpendicularly on the layer, the fraction 

 of these which (whatever their common speed) pass without collision will 

 obviously be only 



1 — n-^Tr.^Sx ; 



for two spheres must collide if the least distance between their centres is not 

 greater than the sum of their radii. It is, of course, tacitly understood when 

 we make such a statement that the spheres in the very thin layer are so scattered 

 that no one prevents another from doing its full duty in arresting those which 

 attempt to pass. Thus the fraction above written must be considered as 

 differing very little from unity. In fact, if it differ much from unity, this 

 consideration shows that the estimate of the number arrested will necessarily 

 be exaggerated. Another consideration, which should also be taken into 

 account is that, in consequence of the finite (though very small) diameter of the 

 spheres, those whose centres are not in the layer, but within one diameter of it, 

 act as if they were, in part, in the layer. But the corrections due to these 

 considerations can be introduced at a later stage of the investigation. 



7. If the spheres impinge obliquely on the layer, we must substitute for hx 

 the thickness of the layer in the direction of their motion. 



If the particles in the layer be all moving with a common velocity parallel to 

 the layer, we must substitute for Sx the thickness of the layer in the direction 

 of the relative velocity. 



If the particles in the layer be moving with a common velocity inclined at 

 an angle ^ — 6 to the plane of the layer, and the others impinge perpendicularly 

 to the layer, the result will be the same as if the thickness of the layer were 

 reduced in the ratio of sin 6 : 1, and it were turned so as to make an angle 6 

 with the direction of motion of the impinging particles. 



8. Now suppose the particles in the layer to be moving with common speed 

 v u but in directions uniformly distributed in space. Those whose directions of 

 motion are inclined at angles between (3 and /3 + dfi to that of the impinging 



particles are, in number, 



?i 2 sin /3^/3/2 ; 



and, by what has just been said, if v be the common speed of the impinging 



