354 BELL SYSTEM TECHNICAL JOURNAL 



will denote the corresponding value of T, whence Tc is given in terms of 

 Re and b by (3.22). 



A formula for dP{R)/dR could of course be obtained directly from (3.4) 

 but it will be found preferable to obtain it indirectly from the less cumber- 

 some formula (3.8) containing the auxiliary variable T defined by (3.6). 

 Evidently, since b does not depend on R, 



dP{R) ^ dPjR) dT_ ^ 2bR dP{R) 

 dR dT dR 1 - b'- dT ' ^ ^ 



Thus, since the factor IbR/il — b") cannot vanish for any value of R (except 

 R = 0), the only critical value of R must be that corresponding to the value 

 of T at which dP{R)'/dT vanishes, namely Tc, since Tc has been defined 

 to be the value of T corresponding to Re- (Incidentally, equation (Bl) 

 shows that Tc is equal to the value of T at which P(R) is an extremum 

 when P(R) is regarded as a function of T.) From (3.22), 



Rl Tc (32) 



1 - b' b 



Evidently Tc and Re must ultimately be functions of only b. The next 

 paragraph deals with Tc, which evidently has to be known before Re can 

 be evaluated. 



From (3.8) it is found that, since dh{T)/dT = I\{T), 



= nm -^ + 



r_L , h{T) 1 



(B3) 



'12T h{T) b_ 



Hence, since P(i?) does not vanish for any value of R (except R = Q and 

 R = oo), Tc will be a root of the conditional equation obtained by equating 

 to zero the expression in brackets in (B3). This conditional equation is 

 transcendental in Te and apparently has no closed form of explicit solution 

 for Tc ; and its solution by successive approximation, or otherwise, would 

 likely be rather slow and laborious. However, the bracket expression in 

 (B3) shows that b can be immediately expressed explicitly in terms of Te 

 by the equation 



^ ^ 1 + 2Teh{Tc)/h{Tc) ' ^^^^ 



For some purposes, the following two equations, each equivalent to (B4), 



will be found more convenient: 



T- 2 + ^^/727)' ^^^^ 



l£ = IZ? (B6) 



b 1 - bh{Te)/h{Te) ^ ^ 



