602 BELL SYSTEM TECHNICAL JOURNAL 



also the b, C, and the constant, a, of the recombination function: 

 ■ia'i 



2 — , -\- a 







(78) 



2al + 



C — 3ai 

 Wb + 1 



, . ^ + 1 2 



did-i -r 2. — i — ai 



as = 



6 + 1 [6 + 1 , . 1 



C - 4ci 



The series in the current parameter are series in ascending powers of the 

 reciprocal of C Writing, for convenience, 



(79) 7 ^ 1/C, 



the differential equation (27) may be put in the form, 



b+ 1 



7 [1 + 2P] 



(80) 



1 + 



P 



GG' 



, b- 1^2 



+ 7 — 1— G 



G - yP 



1 + '-4-' p 



R = 0, 



using the prime to denote differentiation with respect to P. Consider 

 expansions of the form, 



(81) G = Z Ajy\ 



J=JO 



in which the -4's are functions of P to be determined. Substituting in the 

 differential equation, there results 



(82) 



j=JO "I^JO l_ 



[1 + 2P] 



00 



- E An' 







1 a' 1 ^ ~ 1 i I 



.4y.4^ H j— .4,- J, 



;■+'«+! 



L ^ 



7?7 = 0. 



Since the expansions are to hold for arbitrary values of 7, the .4's must, 

 for the cases of field opposing and field aiding, for which the solutions 

 pass through the {P, G)-origin, vanish identically for P equal to zero, 

 and be determined by equating to zero the coefficients of given powers of 

 7 in (82). It can, without loss of generality, be assumed that the coefficient 

 of the leading term in the expansion, Aj^ , is not identically zero. Then, 

 from (82), it is found that there is no expansion for jo = 0, that is, no 

 expansion starting with a term independent of 7. Formal expansions can be 

 obtained, however, foryo = — 1 and for 7*0 = + 1. These may be identified. 



