356 BELL SYSTEM TECHNICAL JOURNAL 



be foun d by substituting Tc into (B4). Fourth, from Tc/b the value of 

 \/Tc/b is found, and thereby the value of Rc/y/l — b"^ and thence Re . 

 Finally, Max. P{R;b) is computed by inserting the critical values into any 

 of the various (equivalent) formulas for PiR;b), namely (3.4), (3.7), (3.8), 

 (3.10) or (3.12). 



APPENDIX C 



FOMULAS OF THE CURVES IN FiG. 4.3 



The first six equations of this appendix are given without derivation 

 and almost without any comments because they correspond exactly and 

 simply to the first six equations, respectively, of Appendix B. Beginning 

 with the second paragraph of the present appendix, the close correspondence 

 ceases. 



dP(r) _ dP{r) dT _ -2b dP(r) 



dr dT dr (1 - ^2);^ dT 



(1 - bVc ~ T • 



dP(r) 

 dT 



(CI) 

 (C2) 



= P(r) ^ + 



[l+b^- 1] (C3) 



12T ^ h{T) b\ ' ^^^^ 



* 3 + 2r, h(T,)/Io(Tc) ' ^^^^ 



b 2 Io{Pc) 



Tl = 3/2 



b 1 - bh{Tc)/Io(Tc) ' ^"-"^ 



The bracketed expression in (C3) is seen to be obtainable from that in (B3) 

 by merely changing T to T/3 wherever T does not occur as the argument 

 of a function; hence the three equations following (C3) are obtainable from 

 the three equations following (B3) by correspondingly changing Tc to 

 Tc/S. (In this appendix, as in Section 4, small c is purposely used as a 

 subscript to indicate a 'critical' value, whereas in Section 3 and in Appendix 

 B, capital C is used for that purpose.) 

 For use below, it will here be noted that 



h{Tc)/h(Tc) = N,{Tc)/No{Tc), (C7) 



as will be seen by dividing (3.16) by (3.13). On account of (3.17) and (3.14), 

 (C7) shows that for large values of Tc the right side of (C7) is equal to 1 

 as a first approximation, and to 1 — 1/2 Tc as a second approximation; 

 thus, for large Tc, 



