418 BELL SYSTEM TECHNICAL JOURNAL 



By use of the parametric equatiuns (4) and (5), Fig. 2 has been ])lotted 

 for r = ^.S^ (TE 01» modes) and values of n from 1 to 9. This drawing 

 shows that, for each discrete value of r, minimum IT P is given by points 

 on the "envelope" of the family of curves. 



The standard method of obtaining the envelope is to express If as a 

 function of /' with )i as parameter (r is assumed fixed, for the moment), 



■J 7,' 



i.e., ir = F(P, //),an(l then set — = 0. However, in this case it is easier 



dn 



to express IT = G(P, <p) and (p = H{ti), whence 



dF ^dG d^ 

 dn dtp dn 



fir" fi /» 



and the envelope is obtained by setting r- = provided t- 5^ 0. We 



d<p on 



proceed, therefore, as follows. 



Assume p is continuous, and solve (4) for p, obtaining; 



sin^ tfi 

 2^ - cos ^ 



Now substitute (11) in (5). This gives TT' as a function of P and (p'. 



3 



47r- 



sm- (p 



cos ^ \ j-~p - cos^ ip 



(12) 



rJll 



To solve — = 0, we dilTerentiate and simplifv. This yields 

 dip 



5 cos (^ — 3 cos"* tp = — - . (13) 



irP 



Substituting (13) back into (11) yields 



2 sin <p 



P = ^ 



3 cos^ ip 



(14) 



The situation so far is that, with P and r assigned, W lies on the en- 

 velope and is a minimum when v? satisfies (13); p is then given by (14). 

 Obviously, for (13) to hold, it is necessary that 



2-^<> 



'•'() obtain the best value of ;-, the ])rocedure is to differentiate ir„n„ with 

 respect to r, assuming now that r is continuous, and examine for a mini- 



