34 BELL SYSTEM TECHNICAL JOURNAL 



of equations is equivalent to the more convenient equations (11-C), 



T 4 [SP 



F 1+* \ k 



(11-C) 



Thus the ratio of T to F is fixed as soon as either / or P {)/ k is fixed. 

 It will be convenient to adopt vP /2k as the arbitrary quantity and 

 to denote it by D, so that 



D = y/Po/2k, (12-C) 



whence P = 2D 2 k, (13-C) 



and t = 7)~ ] ' (14 " C) 



and T = iDF. (15-C) 



Since only positive values of / and D are physically admissible, equa- 

 tion (14-C) shows that the admissible range of D is to 1. 



From (13-C), (14-C), (15-C) and the defining equation F = wL/R 

 the two sets of equations (32), (33), (34) and (35), (36), (37) follow 

 readily from the two sets of defining equations (3-C), (4-C), (5-C) and 

 (6-C), (7-C), (8-C), respectively. 



The formula for plotting the curves in Fig. 8 depends on the exact 

 equation for J/Po which is 



p i + r 2 r(i + r 2 ) ' K } 



By substituting herein the values of P , /, and T expressed by (13-C), 

 (14-C), (15-C) the equation for J/k becomes 



1= 2D ' A + 16D 2 F 2 -D 



k~l + WD 2 F 2 ^F(1 + 16D 2 F 2 )' ( ' 



which is thus the exact formula for the relative impedance J, k of 

 each of the excess-simulators in Fig. 7 when these are proportioned in 

 accordance with the formulas (32), . . . (37). 



A semi-graphical method will now be outlined in the remainder of 

 this Appendix. In this method the ratio T/f is of frequent occur- 

 rence and will be denoted by d. Then, recalling that P-\-iQ = J, it 

 will be seen from equation (1(J-C) that P P» depends only on/ and d; 

 while Q/Po depends on /, d, and /. These observations are the basis 

 for the method now to be described for evaluating the three para- 



