NETII'OR'KS CtiXI.IIXIXd VIIV) Rr.lCT.IXCf'S 3S0 



p. f e i4+i4sjZi+/ljjZj+i4jj.3»ZjZj ,„. 



1 luriforf i = -J — p-j „ , . y , . ^-^ (>i) 



when- .1 is tin.' iliscrimiiiant of the resistance network alone and /I/,.**.// 

 denotes the cofactor of the product of the elements of A located at 

 the intersections of rows /, k and / with columns^, k and /, respectively. 

 I"or con\eiiience this is written as 



a + bZ2+cZ3+dZiZ3 ,.. 



^ ~ a, + biZi + CiZ3+diZiZ{ ^ ' 



The constants of (3) and (4) are real and positive since they are 

 cofactors of terms in the leadint; diagonal of the discriminant of a 

 resistance network. The determinant liciii^ sxinmclrical, there is 

 the following relation among them : 



(adi — a id + bci — bic)- = i{bd I — bid){aci — (liC) . (5) 



The function to be studied is, then, a rational function of two 

 variables, having positive real coefficients determined by the resist- 

 ances alone. Furthermore, if one reactance is kept constant while 

 the other is varied, the function is bilinear. The particular property 

 of the bilinear function, which has been studied in great detail, of 

 interest here, is that by it circles are transformed into circles.^ 



When, as in this case, the variable in a bilinear function is a pure 

 imaginary, the function may be rewritten in a form which gives 

 directly the analytical data needed. I'"or suppose 



u+vz ,„. 



w = — -— (6) 



Ui + ViZ 



where z is a pure imaginary and the coefficients are complex. This is 

 w=—-\ r— ^— . (7) 



Vi Ui+ViZ 



Multi[)lying the second term by a factor identically unity, 



t' , U — UiV/Vi ^Vi'{Ui + Viz)+Vi{Ui'+Vi'z') ,„, 



■w= 1 i X 7-r — r- • (8) 



I'l «I+t'l2 UiVi -|-Mifl 



where primes indicate conjugates, or 



UVi' + tli'v tlVi-UiV / Ui' + ViZ' \ ,^. 



UiVi' + Ui'vt tllVi' + Ui'l'l \ Ul+ViZ )' 



• G. A. Campbell discusses, in the paper cited, the theorem that if a single element 

 of any network be made to traverse any circle whatsoever, the driving-point im- 

 pedance of the network will also describe a circle. 



