/>A7»7.V(;-/'()/.v/ iMiT.n.ixcr. or rno-Mi.sn cikci'its 677 



i>l ilif i|ii.i<lrati(' r(|iialiun (21), wliicli is rrpiMU-tl .ii ilic hoitnm 

 of till' tal>lo. Then fiiul c li\' sulitractiiiK this \aliii- of il from u;. Xext 

 coinputi- 7"i. 7".. and 7"j, assij;nins siijns so that tho identity b^'^, 

 ■\-hi'l\ + hi'l'3 = is siilisfii'd; this is possil)U' siiuv the e(|iiation for 

 (/ was oblaininl by rationalizing; this relation anions the 7"s. There 

 are. in ijenerai. two sets of siijns for which this identilv- is satisfied; 

 it is iniinaterial wliirh set is chosen since the si^iis of all the 7"s may 

 he chan^;ed without chanj-inj; the values of any of the elements. 

 Then compute I't. I':, and I'.i. assijjnin^ positive \alues to each of 

 these. With the \aliies of all these ciiiantities determined, the values 

 of the elements of the networks can be calculated directly from the 

 formulas >;i\en in the Ixnly of the table. If this solution turns out to 

 be impossible, that is. if the \alue of an element is found to be nega- 

 tive or complex or if the value of a mutual inductance is ft)und to be 

 greater than the sciuare root of the product of the associated self- 

 inductances, it means that the conditions u|)(in tin- mois .iiul poles 

 are not satisfied. If the conditions established in ilu- \\v>[ |),iri of 

 this pajx^r are satisfied, the solution is possible. 



These formulas gi\e all the special cases of ilu' i-k-\i-ii networks 

 automatically, that is. the values of the appropriate elements will 

 turn out to Im.' zero or infinite, as the case ma>' be. Since each of these 

 ele\en networks co\ers the entire domain, the>' are all miitualK' 

 equivalent at all fretiuencies. These are the only networks without 

 superfluous elements which cover the entire domain, that is, any net- 

 work co\ering the entire domain must be one of these eleven or a 

 network obtained from one of these by introducing additional ele- 

 ments. Kach of the eleven contains just se\en elements; thus the 

 prediction that a seven-element network woulfl cover the entire 

 domain is verifiefl. The three remaining networks of this same type, 

 one from Circuit and two from Circuit !> of Table III give special 

 cases only, in the sense that each of these can realize only those im- 

 (K'dances which have a pole lying on Line 2; thus each of these three 

 contains a superfluous element, since all the points on Line 2 can be 

 realized by six-element networks, as shown in llie fourth column of 

 the table. 



Network 1 of Fig. 1 is of particular interest since it consists simply 

 of two branches in parallel, each containing resistance, capacity, and 

 self-inductance, with mutual inductance between them.'" By Theorem 

 II, this network can l>e m.ide ec|uivalent to any network whatsoever 

 obtained from a two-mesh circuit. 



'° If will be shown in a sul>s«|uent paper that any (lriving-|Kiint impfflanie of an 

 Fi-mcsh circuit ran bo rcalizc<l by a network of n branches in fKirallel, each branch 

 containing resistance, capacity, and self-inductance, with mutual inductance between 

 each pair of branches. 



