236 THE BELL SYSTEM TECHNICAL JOURNAL, JANUARY 1953 



Integrations over 5 and <f} again go with (112). Beofore writing down the 

 result we shall pass over to a cylindrical coordinate system defined by 



7ll = y cos 6 7x = 7 sin 



7 dy sin 6 dB = yj. dyi. dyj. 



The result of the first two integrations then reads 



Y =- ^\Mmf" exp [- )SMc' - ^m{c - u'f] 



TT 



j[^ dy\\ exp I - my]\ + 2/3^7,1 (c cos i/' - ^^ ^ (1 - cos x) j 



[ 7-'- dy^ exp [i3ilf7i]-/o(2jSilfc7x sin rP)'U [2/3 ,j^,^ t^7-^ sin x ) 



The first of these two integrals is elementary, the other is Weber's second 

 exponential integral ^ which equals 



[ exp {-vY)Uat)imt dt^^ exp (^) /„ (^) (114) 



This yields for Y exactly the expression (113). The identity (111) is 

 thus proved, and with it the convolution theorem. 



The theorem just proved reduces the velocity distribution for arbitrary 

 field and temperature to two components, one containing the field, but 

 not the temperature, the other the temperature but not the field. In each 

 of these components, in turn, the variable parameter scales out; thus the 

 general distribution reduces to two basic ones one of which is the 

 Maxwellian one: the other is worked out partially in the calculations of 

 the Sections IIC and IID. The special case of heavy ion mass has been 

 published independently by Kihara^^ without any apparent knowledge 

 of this theorem which was available in the literature without complete 

 proof. Kihara's form of the theorem is that heavy ions in a light gas 

 have an off-set Maxwellian distribution, with the gas temperature as 

 parameter if the mean free time condition is obeyed for their collisions. 

 Such a function is indeed the convolution of a Maxwellian distribution 

 and the 5-function discussed in the Sections IIB and IIC. 



The general distribution function resulting from (107) cannot be 

 written down explicitly because this goal was never achieved for h{o). 

 However we do find a result which is almost a full substitute for this, 



*• Watson, G. N., A Treatise on the Theory of Bessel Functions. Cambridge 

 University Press, Section 13.31, 1922. 



" Kihara, Taro, Rev. Mod. Phys., 24, p. 45, 1952. 



