220 BELL SYSTEM TECHNICAL JOURNAL 



from which it appears that a resonance frequency occurs for 



h 



n = — =Wo. 



a 



Since e/m for the electron is — 1.77c X 10', the earth's magnetic field 

 of about 1/2 gauss will produce a resonance frequency at l.-tXlO^ 

 corresponding to a wave length of 214 meters, while all heavier ions 

 have resonance frequencies far outside the spectral region to be 

 considered. 



The assumption that the components of the ionic motion are simple 

 harmonic, in spite of the fact that the motion of the ion is rather com- 

 plicated, is justified as follows. From (1) we find that the velocity 

 of an ion (r), say v, is made up of the complementary solution, V,' 

 and the particular solution Vr"=J{E). The latter depends upon the 

 impressed force E, while the former has constants of integration 

 determined by the position and motion of the ion at the last collision. 

 The complete current is thus 



I^^E+y^eVr' + NefiE). 

 ■iir ■*"' 



The second term, however, a\-erages out over a large number of ions 

 since the initial conditions are random;* hence, as far as the effect 

 upon wave propagation is concerned, we may treat all quantities 

 as periodic. 



Following the usual i)n)cedure for the investigation of the propaga- 

 tion of waves in media of this kind, we shall rewrite equation (2) in 

 terms of the com|K)nents of the electric field, tinis for each t\ pe of imi : 



., <T.\ - - 



47r/, = f 1 -f r .\ X - i -^r-^, y=uX - ia Y, 

 \ 11^- — 11- J iio' — n- 



T.V-" 



Wo — M" V Wo" — KV 



in which =ci, or 3.2X10' for an electnui and o.2 . 10' ,- for 



a .1/ 



an ion of mass ^1/. In order to avoid complicated formulas, the 



summatif)ns which must be carried in equations (3) to lake account 



• It is luTC iis-sunud ih.il the iman lime lntWTcii collisions is large compared to — . 



