1186 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



(P (and noneon*U(/)): 



- div grad (P = (40) 



and 



Ol((P)+i^ = 0. (41) 



By operating with div grad on (41) we obtain 



(R''((P) = 



(we consistently use primes to denote differentiation with respect to the 

 argument of a function of a single variable — e.g., 



whence, 



2(R((P) = A(P + B (42) 



(A, B: arbitrary constants). Substituting (42) into (41) we obtain 



whence 



or 



(P = c{x, y, z)e"^' - B/A (A 9^ 0) (43a) 



(P = c(x, y, z) - Bt {A =Q). (43b) 



From (36) it follows that 



div grad c{x, y, z) = 0, (44) 



that is, c must be harmonic. 



In brief, if (R((P) is of the form given in (42), any V{t) and (43) constitute 

 solutions to the flow equations for any harmonic c{x, y, z). Other forms of 

 (R((P) admit no solutions with V = V{t). 



It is evident that when recombination is absent time variation is also 

 absent, and vice versa. The solutions reduce in this case to: 



"U = C (C: arbitrary constant) (45) 



(P = c{x, y, z). (46) 



