LAMINATED TR.\NSMISSION LINES. II 



Appendix IT 



OPTIMUM PROPORTIONS FOR HEAVILY LOADED CLOGSTON CABLES 



We wish to find the lowest root xi of the equation 



1 Ji{xa)No{xPi) - N'i(xa)Jo(xpi) 

 xpi Ji(xa)Ni(xpi) - Ni{xa)Ji{xpi) 



, J_ Ji(xb)No(xp2) - Ni{xb)Jo(xp2) ^ Mo J P2 

 XP2 JiixpdNiixh) - N.ixpdJiixb) a °^ PI ' 



1203 



(A9) 



where h is fixed and mo/m ^ 1, 'M\d it) minimize this root as a function of 

 a, pi , and po . 



Since we expect xi to approach zero as mo./m approaches infinity, we shall 

 replace the Bessel functions appearing in (A9) by their approximate values 

 for small argument, namely 



Jq{x) ^ 1, 

 Jiyx) ~ 2X, 



Noix) ^ - log 0.8905a:, 



T 



2 



Niix) ^ , 



(AlO) 



for I X I <C 1. Then the equation becomes, approximately, 



1 



1/xia 



+ 



I/X1& 



Xipi 2( — «/Pi + Pi/a) Xip2 hi-pi/b + h/po) 

 which may be solved for xii to yield 



Mo , P2 



= - log - 

 M pi 



2m 



Xi 



Mo log (p2/pi) \_pi — a 



1 



+ 



2 



P2. 



(All) 



(A12) 



By inspection xi will be a minimum, considered as a function of a, when 

 a = 0. Setting a = and then equating to zero the partial derivatives of 

 Xi with respect to pi and po , we get the pair of equations 



1 



+ 



Pi[ log (p2/pi)] |_pi h — P2J pi log (p2/pi) 



P2[ log (p2/pi)]" LpI 



6^ - P2J 



+ 



Zp2 



{b - P2) log (p2/pl) 



= 0, 



= 0, 



(A13) 



