WAVE PROPAGATIUX Ol'ER CONTINUOUSLY LOADED WIRES 205 



If (39) to (42) be solved for U, V, U', V, it can be verified that the 

 solutions are the definitions of these functions already given in equa- 

 tions (10).*^ 



The exact formula for the internal impedance of a wire with con- 

 tiguous sheath has been given in (36). In terms of the functions U 

 and V, this formula becomes 



By using the series for these functions and discarding all terms 

 of degree higher than w", the approximation given in the body of the 

 paper (equation 5) may be obtained. 



APPENDIX B 



\^ hen the recurrence formulas are applied to the Taylor series, it is 

 found that 



T- r'' rW 3 \ t'" / 2 1 2 \ 



^='+2+6T. + 24('-/)-T2o(,v-7) 



^ 2v 6V V-/ 24Vv v^ 



, rW . 7 , 24\ T« /3 X^ , 120\ , ,,., 



These series converge for 



< 1, which condition is satisfied by 



the sheath dimensions of any practical continuously loaded conductor. 

 A considerable number of the terms in the series for U and F are 



^ A relation that can be used to advantage at times is 



V h 



U'V - uv = -■- = --. 



X a 



This relation corresponds to the similar one for the Bessel functions themselves, 

 namely: 



J„'{z)KJz-) - J„{z)K„'(z) =1. 



z 



14 



