660 BLLL SYSTEM TECllXICAL JOURX.-IL 



Provided there is one pair of conjugate complex roots of the numerator 

 of the impedance, ai and a.., the \ahie of d is restricted to the range 

 from zero to the smallest real root of (24), that is, 



0^d<au{ai+a->){a3+ai). (2(i) 



In the case of four real roots, ai>a2^a3^«4, the parameter d is 

 restricted to the values 



()<(/<(/„(a, + a,.)(a.i + a4), 1 



\ (27) 



(/„( (* 1 + ttn ) ( a- + (V I ) < r/ < «i,(Qf 1 + or.,) (ofo + a:i) . ' 



Thus there are, in gener.il. iwci (ii>liiKl ranges for the \alue of d in 

 this case. The corresponding tloniain of \alucs of the rt)ots and poles 

 consists of two non-connected regions, so thai it is impossible to pass 

 by a continuous transformation from a set of \alues in one region to 

 a set in the other. 



Formulas (3) ((>) are symmetrical in three tlifferenl respects, since 

 they remain unaltered upon interchanging certain pairs of elements, 

 which maN' he any one of the three following sets: 



(a) bt and />■.., (Z,, .uui </, a-.t and (</■.. — </), 



(h) hi and ft.i, «„ and ((4, fli and Us, 1- (28) 



(c) b2 and b.i. a, and (/, «i and {a< — d). ' 



These three sets corres|)ond to interchanging resistances and in- 

 ductances, inductances and rapacities, ant! resist. mces ami capacities, 

 respectively. 



Since d is always positi\e or zero, formulas (4) ((>) lead to simple 

 necessary conditions, nameU', 



,i:,b:i-aib«>{), (2!t) 



-a,hr + a..bib:t-aJ)3-^0, (30) 



aibi-a^-i>0. (31) 



The first anci third of these conditions are con\eniently interpreted 

 in terms of the roots and |)oles: the simi of the reciprocals of the poles 

 is aigebraicalh' greater tli.m or t.-(\u,\\ to tin- sum of the reciprocals 

 of the roots; and the sum of tin- poles is algehraiially greater than 

 or ef|ual to the sum of liu- roots. 



3. Domain ok Poi.ics iok .XssK.NEn Roots 



The conditions (2) ((i) define a dom.iin of values for the roots .ind 

 poles without distinguishing in any way those roots and poles which 

 may be <lio--cn independently. |-"or many purposes it is convenient 



