TRANSMISSION OVER SUBMARINE CABLES 111 



the American Telephone and Telegraph Company upon the Victoria- 

 Vancouver submarine cable. The calculated values were obtained 

 by both the approximate and the exact methods, discussed in the 

 preceding pages, in which the armor of the cable is treated, respec- 

 tively, as a continuous sheath and as a ring of wires. The modifica- 

 tions which must be introduced to include the effect of the conduct- 

 ing tape are outlined in the discussion of the general theory. The 

 agreement between the calculated and the measured values of return 

 resistance proves that the method developed in the present paper 

 is accurate even at the highest frequencies employed in telephony. 



Note I — Note on Bessel Functions 



The Bessel Functions of zero order of the first and second kinds, 

 Jo (p) and Ko (p), used in the preceding work are all to a complex 

 argument p = ^'gvTwhere 5 is a real number and i = V- 1. The 

 following formulas^ may be used for determining the values of these 

 functions: 



q < 0.1 



Jo{p) = 1 J'o{p) = — h P 



Ko{p) = log, — = .11593 - \ogeq - -r 



yp 4 



Ko' (p) = - - 

 p 



{Jahnke u. Emde, " Funktionentafeln," pp. 97, 98.) 



0.1 < g < 10 



The reports of the British Association for 1912 and 1915 give the 

 values in this range of the functions ber q, ber' q, bei q, bei' q, ker g, 

 ker'q, kei q, kei'q which are defined by the relations 



Jo (iqv^i) = ber q -\- i bei q, 



iy/i Jo {iq\/i) = her'q + i he'x'q, 



Ko {iqy/i) = ker q -\- i kei q, 



i\/i Ko{iq\/i) = ker'^ -f i kei'q. 



» It is to be noted that this approximation for Ko (p) differs from the expression 

 used by J. J. Thomson, "Recent Researches in Electricity and Magnetism," p. 263. 

 Thomson's formula (2) from which his approximation was derived, contains a 

 number of errors and should read 



Ko (x) = (-C + \og2i - log x) Jo (x) -2J,{x) -\ J, {x) ^\J^){x) 



where C = .5772 log = log 7. 



