GUIDKD-WAVE PHOPAdATiOX TIlKorcill (I YK().MA( iXETIC MKDIA ()03 



Fig. 6 — Qualitative behavior of G{\, a) at large distances from the origin as a 

 function of arc tan <r/X. ro is about 2. 



/J ^2 

 T tends to zero 

 1 — crX 



SO that G tends to zero. 



To complete the picture of the G^-function given by the form and posi- 

 tion of the and / curves it is necessary to see how it behaves at large 

 distances from the origin. This is indicated in Fig. 5 and also by Fig. 6. 

 The latter shows the value of G at large distances as a function of direc- 

 tion. In general, along the line a = cK -\- d {c finite), G will tend to — c 

 for all d. For c = n/jn (which is the slope of the asymptotes to the 7„ 

 curves), G again tends to a constant. Now, however, the constant depends 

 upon d and assumes all values from — « to -f oo as a function of d. In 

 the first ciuadrant the sign of variation of the hmiting value of G with 

 direction c is opposite to that of its variation with d near c = ro'/jn. 

 Consecjuently local maxima and minima arise as a function of direction 

 between successive /^-curves. This suggests the existence of saddle points, 

 which may be verified directly. In the third quadrant, the dependence of 

 G upon c and d does not give such maxima and minima, and indeed no 

 saddle points are found there. Finally it is necessary to consider the be- 

 havior of G as 0- tends to infinity, while X remains finite, corresponding to 

 (1/c) — > 0. If X remains fi.xed, then for X> — l,(7-^T^asa--^±cc; 

 and for X < — 1, G' — > ± ^ as o- -^ ± oc . As X ^- 0, the curves of constant 



G are asymptotic to Xo- = f 1 — -\j — B\, where B goes from — -^ to 



+ X with G. Interleaved with these families of curves are the curves 



