GUIDKD WAVK PROP AOATIOX TIIIJOUOII (nHOMAGNKTIf MEDIA. II 979 



always positive; therefore jS" approaches infinity at h{u/2)/Ii{u/2) — 

 AZ^{u) = 0, and no real vakies of ^ exist thereafter (see Fig. 1Gb). 

 Since this "cut-off" occurs at a finite \'alue of u, the corresponding value 

 of To is zero. This explains the bulging of the corresponding j8 — 7"o curves 

 in Fig. 12(a) to 12(c). 



The next major change in the curves occurs when x exceeds ^, (that 

 is, a exceeds 



(Tl 



V^+M-) 



For p < 2, cTi is still less than 1 — (p/2), so that, initially at anj'' rate, 

 A is still greater than unity. In addition to the infinity of /3^ just dis- 

 cussed, a further infinity arises between u = 0, and the pole of Z^{u), 

 as is seen from Fig. 16(c). jS increases from zero at w = to this infinity, 

 thereafter it is negative, until the pole of ZJ^u) is reached. Thereupon 

 it resumes at |8" = and approaches infinity at the zero of the denomina- 

 tor [/o(i//2)/7i(w/2)] — AZ^{u) already discussed. Thus there are now 

 two branches of the ^^ — u curve; their corresponding traces in the 

 j8 — f plane are shown schematically in Fig. 17. (The computations on 

 which Fig. 12(a) to 12(c) were based were broken off at a = o-i .) 



A further branch is added each time x increases beyond a number of 

 the form (2n + l)/2 (o- increases beyond 



2 



4A + 



V 



2n + 1 



These all resemble the two branches just discussed, until n > 



Fig. 17 — Schematic variation of /3_ with ro for K < X < %• 



