MOTION OF GASEOUS IONS IN STRONG ELECTRIC FIELDS 195 



^r = —irr-, cos;/' —tt-t — cos k + c 



M -\- m M -\- m 



From these equations the value of the Jacobian comes out to be 





sin ^ { cos xp sin k — sin xj/ cos k cos (^ — co) j 



(M + my 



We needs its value only at the position C^ = C^ = C^ = 0. If we take 

 the above equations for C^ , Cy, , Cj- and multiply them respectively by 

 cos K cos CO, cos K sin oj, — sin k, add and set C = we get the identity 



My 



M -i- m 



cos if/ sin K — sin >p cos k cos {(p — oo)] = c sin k 



The curly bracket is exactly the one occurring in the Jacobian which 

 therefore reduces to 



, — 12— ±L = c c sm ^ sm k 



L ^('c,^,^) Jc'=o {M + mY 



and hence 



7' dy d% dQy = ^^,;^ ^^ dC' ^c' ^co 

 ' ^ ^ Mmc 



Substituting finally this expression into (32) and (31) we get the Boltz- 

 mann equation in the form 



dh{c) . 1 , , . 

 dCz t{c) 



M^ (34) 



AirMmc Jc r{c) Jo 



The equation is in need of additional elucidation as regards the exact 

 meaning of c' as a vector and as regards the auxiliary variable x- As to 

 the first point we may describe the integration as occurring over a sur- 

 face in velocity space. This surface is obtained from the relation 



C = c' = Y (35) 



which substituted into (33) becomes 



(M + m)c - my' = My (36) 



