1178 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



and ao has a minimum when 



e = 2/3. (500) 



The minimum value of the low-frequency attenuation constant, which 

 we may call aoo , is just 



«oo 



3V3 



Writing 



3V3 



we find that equation (49(5) takes the form 



J m 



V3 Xp 



" 3V3 (1 - ef ar. 



26 aoc 



(501) 



(502) 



(503) 



for any value of 6. From equation (497), fm is a maximum when satis- 

 fies 



6(2 - 6) 



8 aoo 



(1 - 6)^ 3 V3 «„. ' 

 which is equivalent to the quartic equation 



64aoo 



- W + W -f 



27c 



(0 - 1) = 0. 



(504) 



(505) 



If dm is the root of (505) which lies between zero and one, then the 

 corresponding value of fm is 



Jm — 



V3 



Xp 



'2 - 30. 



irHiQitidm [_2 — dm _ 



(506) 



We observe from either (503) or (506) that fm is inversely proportional 

 to ti . 



The values of dm and C[(2 - 3dm)/{2 - dm)f are plotted in Fig. 

 21 against am/aoo , which is just the ratio of the maximum attenuation 

 constant to the minimum low-frequency attenuation constant which 

 can be achieved with a Clogston cable of the same diameter. When 

 am/ocoo is unity, then dm = 2/3 and fm is zero to the present approximation 

 (a better estimate of fm would be the critical frequency fi defined by 

 equation (467)). For values of am/aoo greater than about five, dm is 



