i>Hiri.\\:-r(>ixi imi'id.ixci-. oi- iuomesii cihcuiis «)7 



\'hv tilth set III roots (£i, E^) has a domain i>f iln' siiim- i\|>e as thr 

 torrespondinjt set of roots on FiR. 3c, since this st-t. being on the rsal 

 axis, is a double set. The sixth rurve f-' is the boimdarN' of ih«' domain 

 lor four real roots so chosen that FiF3 = E\^ and /■';/''i«= Er. This is 

 the siime t>-pe of domain as will l>e descriJKid later under l''in. ."). The 

 curves of I-'ig. 'M are all tangent to the \ertical axis at the same point 

 o; for each of these sets of roots the distance Oa is the geometrical 

 inean \alue of all four roots. 



The general case of four complex roots is illustrated by Fig. 4 for 

 the numerical values ai=— l+i'lO, ««=— 1— I'lO, 03=— 2 + tl5, 

 014= — 2 — I'lo. I'or all complex roots the poles must also lie complex; 

 the pole with positive imaginary part must lie in the region boimded 

 by the curve r = ri + r2 + . . . . + r7. This curve is tangent to the 

 vertical axis at the point a, and tangent to a vertical line at the left 

 at the point d. The largest absolute value of any point in the domain 

 occurs at the point c, and the smallest at /; these two points are the 

 jx)ints of tangency of the curve T with circles about the firigin as 

 center. The cur\e F is tangent at the point c to a circle through 

 the origin having its center on the real axis. The coordinates of 

 these points are all given in Table V. 



The general case of four real roots is illustrated by Fig. .5 for the 

 numerical values ni= — 1, «•=— 2, 03=— n, a4=— 7. The domain 

 of complex fjolcs is boimded by the cur\e F, with the critical points 

 defined and labeled as in Fig. 4. The domain of complex poles is 

 bounded in part by two segments on the real axis, one lying in the 

 interval between a\ and ai. the other between «j and «<. Approxi- 

 mately, these segments are from —1.13 to —1.93 and from — .').13 

 to —6.70, for this numerical example. The points on these segments 

 are in the domain of poles, corresponding to double real poles. The 

 domain of real poles is shown by the graph below the axis, each point 

 of this graph representing two real values, the two points on the real 

 axis reached by following the ±4.5° lines through the point. The 

 domain of real poles is bounded by the continuation of the curse F 

 and the tangent lines corresponding to the four roots. This gives 

 two corners associated with the two segments on the real axis, and 

 an isolated rectangle. Corresponding to the points in the rectangle, 

 one pole may be chf)sen an>where in the range from ai to «;, and 

 the second pole anywhere in the range from aj to at. Both pfiles 

 inay be chosen in the range from ai to 02, or in the range from a.i to 

 ««, with certain restrictions as shown by the figure, since the curve F 

 cuts oflP the points of the triangles. The two corners and the rectangle 

 are shown by Fig. 5a on a larger scale, with greater accuracy. 



