nRiri.wi-i'oi.xi iMrii).i\cr. or tu'o-mhsii hkiiiis us.t 



I'luin subslitiilinK X = //>, imilliplying nunu'ralor and (Iciinmiii.ildr 

 by X'. anrl dropping llu- siilxripl </, forninla (11) bccomo 



_ aoX«« +o^X«- - ' +a,X2- -' + . . .+a2, -iX+Q2, 



•^" 6,X2--'+/'..X2''-2 + . . .+62, ,X ' '■*"'^' 



whirh may l>e taken as the most general form of a (lrivinK-p<>'"t 

 im|Kdance. This formula, therefore, gives the impedanre of the 

 ririuit for any electrical oscillatif)ns of the form /', where X m.i\ 

 have any value, real or complex. I'ormula (42) may In; written in the 

 alternative form 



(X-a,)(X-a.)(X-a3) . . . (X-gg-QCX-azJ 

 ^-" X(X-^,)(X-/3,) . . . (X-^2,-.) ^'^•^> 



Thus there are '2n roots of .V, rej^arded as a function of X, whicli are 

 the 2h resonant points of the circuit. There are also 2« poles of S, 

 which are the 2m anti-resonant points of the circuit, namely, zero, 

 infinity, and the 2h— 2 resonant points of the circuit obtained b>- 

 op)ening the branch in which the driving-point impedance is measured. 



Upon setting «=2 in ec|uations (43) and (42), formulas (la) and 

 (lb) are obtained, respectively. 



From the fact that the coefficients Ljk.Rjt, and I, Qk satisfy the 

 quadratic form conditions (39), it can be shown mathematically 

 that the coefficients an. «i, . . . , Ozn of (42) are all positive and that 

 the roots ai, a;, . . . , a;„ of (43) have negative real parts. '^ This 

 can also he shown from the fact that the free oscillations of the circuit 



are of the forms /'', /"' f*'"'. Thus the roots occur in pairs 



each of which has negati%e real values or conjugate complex \alues 

 with negative real parts. 



The siime restrictions hold for the coefficients 61, b-: /)■.•„ 1 



and the poles ^:, /3j fiin-i since the denominator of S, with the 



exception of the factor X", is also the fliscriniiiiaiii of a circuit. Thus 

 the general restrictions (2) are obtained. 



In order to obtain the necessary and sufficient conditions that a 

 function of the type (lb) represent a dri\ing-point impedance realiz- 

 able by a two-mesh circuit, set this function equal to the impedance 

 of the most general two-mesh circuit and investigate the conditions 

 which must hold upon the coefficients in order that the two forms 

 may be equivalent. 



" The mathematical work is identical with the mathematics ot the corres|K)ncliiiK 

 dynamical problem. A detailed proof is given by A. G. Webster, /of. cit. 



