424 BELL SYSTEM TECHNICAL JOURNAL 



and the equi-er curves of Z, and the equi-;- and the equi-a' cur\'es of Z' . 

 The equi-r and the equi-cr curves constitute a curvilinear network 

 superposed on the rectangular background of Z = X+iY\ for any 

 assigned pair of values of r and c the value of Z can be obtained by 

 finding the intersection of those particular curves of r and a, and at 

 that point reading of? the value of Z on the rectangular background. 

 Similarly for the evaluation of Z' by means of the network of equi-r 

 and equi-o-' curves. 



For the ff'-range and the a-range contemplated in Fig. 2 — namely, 

 0<ff'<l and 0<(r<l — the Z'-realm and the Z-realm are distinct; 

 their mutual boundary (drawn dashed) is the unit semi-circle, that is, 

 the semi-circle of unit radius hav^ing its center at the origin. The 

 Z'-realm is the region inside; the Z-realm is all the region outside, 

 extending to infinity in all directions through the positive real half of 

 the complex-plane. 



If the ranges of a' and a are extended to include values exceeding 

 unity, the Z'-realm and the Z-realm will cease to be distinct but will 

 overlap. The Z'-realm will expand upward, beyond the unit semi- 

 circle, and ultimately will fill the region of unit width extending 

 upward to infinity; the Z-realm will expand into and ultimately will 

 fill the lower half of the unit semi-circle. Hence for values of a' and a 

 exceeding unity it is preferable to employ individual charts in repre- 

 senting Z' and Z. 



In the language of tuiutii)n-theor>' it ma>- be said that, when a' = a, 

 the Z'-realm and the Z-realm are inverse realms with respect to the 

 unit semi-circle. The straight lines and the circles are inverse curves; 

 the ellipses, and the curves characterized by the equation (A"--(- F^)^ — 

 X"^— Y^/{2a— l)'- = are also inverse curves. 



For r = it is seen that Z' = Z= 1 for all \alues of a' and a. 



For values of r equal to or greater than unity, Z' and Z are pure 

 imaginary, for all values of a' and a. For r = l, Z' lies somewhere on 

 that part of the imaginary axis constituting the vertical diameter 

 of the unit semi-circle, its position thereon depending on the par- 

 ticular value of a' contemplated; while Z lies somewhere on the 

 remainder of the imaginary axis. When r approaches infinity, Z' 

 approaches infinity and Z approaches zero, along the imaginary axis. 



Another graphical method of representing the relative impedances 

 Z = X-{-iY and Z' =X'+iY', based on equations (4) and (5), is by 

 means of the Cartesian curves of the components X, Y and X', Y', 

 with the relative frequency r taken as the independent \ariable and 

 the relative termination {a or o-') as the parameter. 



