676 IU:U. SVSTliM TI.CIISICAL JOURSAl. 



These critical lines and jjoiius arc illustrated, for luniiciical cases, 

 by Figs. 4 and o. The graph showing the domain of real poles in 

 Fig. 5 is inaccurate to the extent that the critical line> ii.ixe liecn 

 spread somewhat ai)art from each other in order to show the seciuence 

 in which they occur. The actual cur\'es are shown accuraleK' drawn 

 and on a larger scale in Fig. Sa. K\-en on this scale. ("ur\e 2 cannot 

 be distinguished from the side of the rectangle. 



The diagrams for the domain of complex poles, as illustratetl 1)\- 

 Figs. 4 and n, are approximately symmetrical with respect to the inter- 

 changing of inductances and capacities, with corresponding inter- 

 changes in all the cur\es and formulas. Thus b and g correspond, 

 r antl /. (/ and c, 2 and .3, •") and (i, S and 9, ai and aj, a-i and as; while 

 II. 1. 4, 7, and 1(1 remain unchanged. In ilie domain of real poles shown 

 by Fig. o, this symmetry does not appear. The explanation of this 

 apparent discrepancy is as follows: Upon interchanging inductances 

 and capacities, the \alues of the roots are changed to their reciprocals. 

 Thus Fig. .") is symmetrical with the corresponding figure drawn for 

 the case of roots equal to —1, —1 2, —1,5, and —1 7, and thus 

 symmetrical with the figure drawn for roots at —7, —7 2. —7 ."). 

 and —1, since the relati\e distribution of the roots is the same. This 

 set of roots differs not very considerably from the original set of 

 roots, in reverse order. In the main, therefore, the two figures ma\' 

 be expected to be approxiinatiK the same, that is, the original figure 

 symmetrical with itself. In the rectangle, however, very small 

 numerical changes in the constants make relatively large changes in 

 the curves; so it is not surprising to iind a lack of symmetry here. 

 If the roots are assigned so tli.it the prodint of two roots is equal to 

 the product of the other two. there will be true s\-mnu'tr\- in the 

 corresponding diagram. 



Table III li>ts :58 circuits, gi\iiii4 .i tni.il nf 102 netwuik^. Of these 

 networks, three are essentialK' thi' (.'(niix aleiii nl lutwink^ nbt. lined 

 horn a one-mesh circuit, one realizes oiiK tlio^e imped. mc^■^~ which 

 have one pair of pure imaginary roots, and. of the '.IS remaining. II 

 have superfluous elements. This leavis .i tot.il of ■")7 networks, oi 

 which 11 realize the entire doiuain asgi\cii li> riumiin II, 12 re.tli/e 

 regions in the domain as gi\en by Theorem 111. 2M re.ili/e critic.il lino 

 in the domain, and 11 realize critical points. 



The ele\en networks of Theorem II are iiK hided in the In^t colinmi 

 of Table III and shown in detail by Fig. I. I'(irniiil.i> lor tlu' com- 

 putation of their elements are gi\en 1)\- T.ible 1. rini> the \,ilues 

 of these elements can be comjiuted direct l\ in teini> nl the nuttii ients 

 of the imiH'dance expression as slated in i he lorni ( Ibi. The follow in. i; 

 method of computation is convenient: I'irst coinimtt' d ,is the root 



