1196 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



cally in ni. Making use of the algebraic lemma that 



A + Bx +- + (D+-)ln 

 X \ xj 



- 1 



^ 



implies A=B = C = D = E = 0, we arrive at two possibilities : 

 Possibility 1: 



{q{t) = r{t) = and j - y = Kr"^') 



(K, L: arbitrary constants) 

 This yields from (92), (89a) and (91): 



2kT 



eN 



(R(ni) = UM 



ai(5C,0 = KC'-'K ^ e"-' (w(/) - 5C) 



and 



5C(a;, y, z, = € ''' s{x, y, z) + e' 



/•"[ 



Lm{t) -\~ m'{t) 



(93) 

 (94) 



dt (95) 



where s{x, y, z) is any harmonic function. 

 Possibility J 2: 



m = M,q(t) = Q,r{l) = R) 



(M, Q, R : arbitrary constants) 

 This yields from (92), (89a) and (91): 



eN ni 



(" 



+ 



M 



ni + (3in 



'U 



M 



^(5C, /) = (M - 7) A 



r w(/) - 5C "| 

 L M-y J 



and 



^(x, y, z, /) 



QtUM-y) 



u{x, y, z) 



+ e^'^^^-^> /< 



-(QtKM-y)) 



[r + »»'« + M^ -w] 



(i/ 



(96) 



(97) 



(98) 



where u(xj y, z) is any harmonic function. 



In the absence of recombination, Possibilities 1 and 2 lead to the same 

 result: Equation (97) and 



X,{x, y, z, /) = u{x, y, z) + m(f). 



(99) 



