972 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



magnitude of j8, and it is chiefly for this reason that further progress is 

 possible. For large /3, equation (35) can be written 



iS' = /3o' cot' ^ 



/i(| /3 I ro) 61 Ki(| 13 I ro) 

 ToOlko) 6oKo(|^|ro) 



/o(| ^ I ro) 



1 PF,.o(2|^|n)) • (37) 



/i(l /3 ko) M. .2^ _ j^ T^x.o(2 I /3 I ro) 

 Mo 



where x is now given by equation (36). Equation (37) is now solved by 



-7 



-6 



-5 



O 



^5. -3-r: 



-2 



•^ 1 



O 



u 

 o 

 Q. 2 



0.2 



0.4 



1.0 



1.2 



1.4 



1.6 



0.6 0.8 



ToyGo COT Tp 



Fig. 12 ■ — Reduced forward and reverse propagation constants versus reduced 

 radius of a cylindrical heli.x (loss-free case) for various a and p. The range en 

 < (7 < (To where 



= /jA+ 



pz 



and 



/' + M 



contains an infinity of shape resonances and is not shown here, (a), above, p = 0.2. 

 (b) p = 0.6. (c) V = 1.0. 



