RECENT STATISTICAL THEORIES 731 



It will be observed that deflections smaller than 90° are equally 

 numerous with deflections greater than 90°, so that on the average 

 the electrons after impact have no reminiscence of their prior direction 

 of motion, as I mentioned earlier. Also, comparing (96) with (94), 

 one derives the expression for mean free path, 



/ = 1INtR\ ■ (98) 



cited already in a footnote. To appreciate the most important 

 feature of the expression (97) we must however return to the velocity- 

 space. 



In the velocity-space, the electrons of which the paths in coordinate- 

 space are deflected through angles between 29 and 2d -}- d2d execute 

 leaps from the compartment C into other compartments of the spherical 

 layer aforesaid, located on a certain region thereof. These occupy 

 a belt or collar on the sphere, intercepted between two cones drawn 

 with their common apex at the centre, their common axis pointing 

 towards C and their apical semi-angles equal to 6 and 6 -\- dd respec- 

 tively. Now the area of this belt is itself proportional to sin Odd. 

 This is very important: for it means that the electrons which are 

 bounced out of C by collisions are sprinkled uniformly over all the 

 rest of the sphere. More yet: it means that the electrons which 

 are bounced out of any cell of the spherical layer are sprinkled uni- 

 formly through all the rest of the layer. 



Consider then the interchange of electrons between two cells of the 

 layer, say C at (^, r;, f) with volume dT, and C at (^', -q', f') with 

 volume dr . The number leaping from C to C is equal to the total 

 number of impacts occurring in C multiplied by the ratio which the 

 volume of C bears to the volume V of the layer. The number 

 leaping from C to C is equal to the total number of impacts occurring 

 in C , multiplied by the ratio which the volume of C bears to the 

 volume of the layer. The excess of the latter over the former is then: 



Z{^', v', ndr'idrlV) - ZU, V, ^)dr(dT'/V), (99) 



which with the aid of (96) and (98) may be written thus: 



^[/•(r, V, n -m,v,nidr'. (loo) 



This is the quantity of which the integral with respect to ^', rj', f', 

 extended over the spherical layer, is equal to {b — a)dT — the net 

 rate at which compartment C gains particles through impacts. 



Making Lorentz' postulate about the form of the distribution- 



