260 BELL SYSTEM TECILXIC.IL JOURNAL 



constructed either by combining, i>i parallel, resonant circuits having 

 impedances of the form iLp-\-{iCp)'\ or by combining, in series, anti- 

 resonant circuits having impedances of the form [/C/J+f/Z, />)""']"'. In 

 more precise form, 



PiPl-p') ■ . . {Pin-2-P') ' ^^> 



where II>0 and 6 = po<pi<p2-^ ... <p2n-i^pin=^ -^ The induct- 

 ances and capacities for the n resonajit circuits are given by the formula. 



i--,^=(^Ka='.' 2«-i). 



(2) 



and the inductances and capacities of the w + 1 anti-resonant circuits 

 are given by the formula, 



^^■=z;^ = (5(#rF))-^-^'=''''''- ■••'"-'• '"^' 



(3) 



U'hirh includes the limiting values, 



iip]pi ■ . . pl„-\ 



Formula (1) may be stated in se^•eraI mutually equivalent forms.* 

 This particular form is the driving-point impedance of the most 

 general symmetrical network in which every branch contains an 

 inductance and a capacity in series, with mutual inductance between 

 each pair of branches. This includes as special cases the dri\in^-point 

 impedances of every other finite resist anceless network. 



'Since the impedance 5 is an oclil fuiulion of the frequency, resonance or anti- 

 resonance for p = P implies resonance or anti-resonance for p=—P. In enumer- 

 ating the resonant and nnti-resonant frequencies it is customary, however, to ex- 

 clude negative \alues of the frequency. Thus, in the present case, we say th.it 



there are n resonant points (pi, p , pu-C> and m + 1 anti-resonant points 



ipo = 0,p2.p, Pu-2.p2n='»). 



* The expression for S given by formula (1) may he written in the mutually equiv- 

 alent forms, 



r ,- ///'?-/>' ) (M-p') ■ • • (pi..-.-p') it' „, r....( />i-^') . ■ . (/'i.-,-/>') -it' 



L-"'^ /.(/'5-/'=)...(/^.,-.-/") J ■""' l-"'^>?-/") . . . (/'?n-,-P')J 



If the constant // and all the p/s of these formulas are restricted to finite values 

 greater than zero, the four cases, obtained by separating the plus and minus ex- 

 ponents, are mutually exclusive, but together they cover the entire field. If ^i is 

 allowed to be zero, either the first or the second pair covers the entire field. I'inally, 

 if in addition p-n-i or />j„ 2 is allowed to become infinite, while IIpi„^\ or ///"L-s 's 

 maintained finite, any one of the four expressions covers the entire field. Some- 

 times one, sometimes another way of covering the field is the more convenient. 

 Formulas (2) and (.3) apply to all of these expressions for 5 provided the p/s include 

 all the resonant points and all the anti-resonant points, respectively. 



