656 HELL SYSTEM 1 ECIIMCAL JOURNAL 



by any network of the form of Fig. 7, provided the elements of the network 

 satisfy the following relations: 



L,'L.' + L/L,' + L/L3' = aok\ (7) 



R,R> + R,R,+R-2R3=dk\ (8) 



D,Di + D,D, + DoD,=aik-, (9) 



L.'+W = b,k\ (10) 



i?2 + i?3 = W-, (11) 



D. + D, = b,k\ (12) 



R;D-,- RiD2= ±k-\-aJK+a-,b;b:,-dbi-yi^, (13) 



D;L/-D,L./ = ±kn-aob,^ + (a.-d)b,by-aibi^''^, (14) 



L!'R,-UR;=±kH-dbr+a,bibn-aob2-y'^, (15) 



where D, = Cr\ Di = Cr\ D^^C^-', (16) 



and Li'=L, + .U,..+ .l/i,+ .l/'..<, (17) 



L2'=L.,+ .U„-J/,,-.U,;,, (18) 



L..,'=La-.Ui.+ .l/,.-.l/.;:<. (19) 



///e positive directions in Fig. 7 all being assigned arbitrarily to Ike right. 

 The signs of (13) (15) are chosen so as to satisfy the identity 



(R2D,-R,D,)(L-/-\-L,')-\-{D,W-D,L,'){Ri + R3) 



+ {L.,'R3-WR.){D, + D,)=0. (20) 



'J'he value of d is given by equation (3), which may be written in the form 



d:'{b-?-Ab^b3)-2d(2aJ)i'^+aib-fi-\-2a»b3^-aJ)xb..-2a«bibi-aib-,bi) 

 -'r\a3-br+(ai^ — \a,>ai)b-r + ai-bi^—2{a-ia:t — 2aiai)bibi—2aia3b\bi 



-2(«ia2-2at,rt::)W'.i]=0. (21) 



The parameter k may have any real value other than zero. 



In tlifse foriiuilas the \alue of k is independent of the impedance, 

 but can l)e chosen so as t<j give i^arliciilar forms of the nctworl-;. If 

 the necessary and sufficient conditions as stated by Theorem i , in- 

 satisfied, the values of the elements given !)>• these formulas are 

 positive or zero, and the \alues of the inductances satisfy the usual 

 restrictions. The formulas of Tables I and II. f(ir example, can all 

 be computed by means of Theorem l\ . 



2. Till-. I)ki\in(.-I'(iin I l\iiii)AN( !■: oi- .\ r\\i>-Mi;Mi CiKiiri 



Previous inxestigations of the two-mesh circuit have been directed, 

 for the most part, toward the determination of the free periods (reso- 



