604 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



(7 = ± 00, which are Xo- = (1 — ^ ) + 0(X^)- More detailed information 



on these matters will be found in Appendix II. 



From the G^-diagram it would be possible to determine pairs of X-values 

 with opposite signs, which, for a definite o--value satisfy the characteristic 

 equation, but, for a given y such pairs would not necessarily satisfy the 

 Polder relation (31b). It is necessary to have a procedure which takes 

 account of the latter systematically. Such a method may be based upon 

 the fact that if, for a and y positive, the Polder relation is solved for Xi 

 in terms of X2 it can be thought of as a rather simple mapping of the 

 whole X2-quadrant upon a part of the Xi-quadrant (Xi > 0). Similarly for 

 0" and p negative there is an analogous mapping of the Xi-quadrant onto 

 the X2- quadrant. 



Considering first the case cr, p > 0, the Polder relation may be written 

 in the forms 



X, = '-±J^ = 1 + l + v^ = r(x,). (36) 



1 — (7X2 cr 1 — (xX-i 



From (3G) it may be seen that the curves X2 = const, transform into a 

 bundle of hyperbolae passing through the intersection of a = 1/Xi and 

 0- = Xi — 7); that is, through Xio , ao , where 



= -p/2 + yl'+l , Ai. = p/2 + |/ 



?+' 



These hyperbolae have the vertical asymptotes Xi = — I/X2 , and intersect 

 0- = at Xi = p — X2 . For a fixed positive a less than o-q , Xi decreases 

 from l/cr to c + p as X2 increases from — =0 to 0, but when 0- is greater 

 than ao , Xi increases from l/cr to o- + p under the same circumstances. 

 Thus the whole X2-quadrant is transformed upon that part of the Xi- 

 quadrant which lies between the hyperbola Xi = l/a and the straight 

 line Xi = o- + p- It follows that points in the Xi-quadrant which are, for 

 a given p, excluded from this region, cannot be the site of acceptable so- 

 lutions of the G-equation. 



Since as has already been stated, the Polder relation is unchanged 

 by the substitution Xi -^ — X2 , X2 ^ — Xi , o- — > —a, and p to — p, it 

 follows that for cf and p negative a similar mapping of the Xi-quadrant 

 upon part of the X2-quadrant takes place. The transforms of the hues 

 Xi = const, and so forth may easily be found by using these substitutions 

 in the formulae already given. 



Reference to Fig. 1(a) and (b) will show that ±0-0 are the values of o- at 

 which ;u reverses sign. Therefore we may expect o-q to play a special role 



