182 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



writing the above differential in the form 



— y(r{y)U{x) sin xdxde = — y(T(y)Uix) dn^ 



Here dQ is meant to represent an integration over a solid angle and the 

 subscript 7, that it is over the solid angle swept out by the vector y- 

 The notation makes use of the fact that the choice of the polar axis is 

 arbitrary in such an integration. The function n(x) is the probability 

 of scattering which equals unity for isotropic scattering. In cases where 

 small angle scattering is infinitely probable the above expression becomes 

 meaningless, strictly speaking, n(x) being a 5-f unction at x = and o- 

 being infinite. However if a quantity such as 1 — cos x is multiplied 

 in, which removes the 5-function then the integration gives a finite num- 

 ber which may be denoted by (o--(l — cos x))- 



IC. FORMAL SURVEY OF THE THEORY 



Under the assumptions stated in Part lA we may describe the motion 

 of ions in a gas by their density in phase space. The change in time of 

 this function is described by a Boltzmann equation^ which, in our 

 notation, reads 



ddjc, r, t) dd{c, r, t) ddjc, r, t) 



dt '^ ' dc '^ ' dr 



(10) 



= £ // {M(C')d{c, r, t) - M{C)d{c, r, t)]y<T{y)llix) d%> dC 



The equation is linear in the unknown function d{c, r, t) ; this is due to 

 neglect of ion-ion collisions, as stated earlier. The negative term on the 

 right hand side actually reduces to a known function of c multiplying 

 rf(c, r, t). The positive term is a genuine integral term; it has been shown 

 by Pidduck that the number of integrations in it can be brought down 

 from five to three; this reduction will not be made use of in the following. 

 If there were no terms on the left hand side of equation (10) then the 

 solution of it would have the equilibrium form 



d{c, r, t) = nm{c) (11) 



where n is a constant. This result is a direct consequence of equation 

 (8) which makes the curly bracket in (10) vanish identically when 

 Maxwell ian functions are inserted. 



'* See Reference 4. 



" Pidduck, F. B., Proc. Lond. Math. Soc, 16, p. 89, 1915. 



