DIFFRACTION OF RADIO WAVES BY A I'AHAROLIC CYI.IXDKR 187 



3. m = (0.54953. . . )p' '^ 0.55p'-'. Fi^^ I 1.1 (I)). This valuo of m 

 nmrk.s the chaii.ti'c in type of patli. Since Im /(/i) = Ini./'(/ii) is satisfied, 

 the paths through Id and d have the same eiiuation [see (11.2)], and 

 there is a chance for a situation like that at Ai in Fig. t 1.1 (h) to occur. 

 The high point of the path I' is at /i, and it goes contiiuialiy downhill 

 on either side of /i although its direction changes sharply by 90° at /«. 

 The point m = 0.55p" is just one point on the boundary between regions 

 in the ///-plane corresponding to various types of paths. The boundary 

 lines are ()l)tained by solving condition (10.11) for in as outlined in 

 Items G and 7 of Section 10. Mapping the boundaiy lines 



Im UiU) - fik)] = 



from the auxiliary ii?-plane (shown in Fig. 10.3) to the ///-plane with the 

 help of /// = /'p"/(cosh iv + 1) gives the boundaries between the regions 



Fig. 11.2— Regions of different types of paths of steepe.st descent, and hence 

 ilitforent types of asymptotic expansions, when z = z'/^p. Points numbered 1, 2, 3, 

 are values of m corresponding to'the paths of P^igs 11.1 (a), (c), (b). Points desig- 

 nated bv 5, 6, 7 correspond to Fig. 11.3(1»>- Points 4, 8, 9 correspond to Fig. 11.5 

 and points 11, 12, 13, 14 to Figs. 11.6 (a), (b), (cj, (d). 



