894 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1952 



proximate values for small argument. From the series given in Dwight 

 813.1, 813.2, 815.1, and 815.2, we have 



loix) ^ 1, 



h{x) ^ ir, 



Koix) ^ -(0.5772 + log ir) = -log 0.8905a;, (39) 



A'i(.r) ^^- -\- ^x log 0.8905a,-, 



X 



for I X I <3C 1, where log represents the natural logarithm. If we put 

 these approximations into (38) and if we suppose that the wall im- 

 pedances are so small that 



I cropiZi(7o)/ 2r?o | « 1, | (Top2Z2{'Yo)/'2rio | « 1, (40) 



we obtain, after a little algebra, 



2 2 2 (To[Zl{yo) / pi + Z2(.yo)/p2] /.,x 



Ko = (To — y = — T J — ^-^ . 141 j 



rjn log (po/pi) 



Now further assuming that 



1 Zi(yo)/pi + Z2{yo)/p2 ^ ^ /^2) 



8 (Tor/o log (p2/pi) 



we get by the binomial theorem 



I Zi{yo)/pi + Z2{yQ)/p2 /.Q^ 



7 = o'o + ^ — -. . , . • 143; 



2i7o log (p2/pi) 



If we formally set ^o = 0, A\e find that the attenuation and phase con- 

 stants of the principal mode in a coaxial line with low-loss walls and no 

 dissipation in the main dielectric are 



Zi(7o)/pl + Z2{yo)/p2 /. .X 



a = lie 7 = Ke ;r — z -. — -— , 144; 



2rjo log (p2/pi) 



o T / I T^ Ziiyo)/pl + Z2( yo)/p2 /,-N 



jS = Im 7 = ojVMofo + Im — , — ^-r . i4o; 



2770 log (P2/Pl) 



As before, these approximations for a and 13 will ultimately break down 

 as the frequency approaches zero, but they will certainly be valid over 

 the frequency range in which we are interested in the present paper. 



8 H. B. Dwight, Tables of Integrals and Other Mathematical Data, Revised Edi- 

 tion, Macmillan, New York, 1947. We shall refer to Dwight for a number of stand- 

 ard series expansions. 



