1142 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



and replacing the modified Bessel functions in (337) and (338) with 

 ordinary Bessel functions according to equations (292) of Section VIII, 

 we obtain 



y ^x Ji{xa)No(xpi) - Ni(xa)Jo(xpi) (^nc,s 



' ~g J,{xa)N,(xP:) - N,(xa)MxPr) ' ^ ^ 



Z =^ '^i(x^)A^o(xp2) - Ni{xb)Mxp2) ,^.„. 



' ~g MxP2)N,ixb) - Nr{xp2)Ji{xb) ' ^ ^ 



Substituting (301), (362), and (363) into (359) and setting e/eo = Mo/m, 

 we get after a little rearrangement, 



_1_ Jiixa)No(xpO - Ni{xa)Jo{xpi) 

 XPi J\ixa)Ni(xpi) - Ni{xa)Ji(xpi) 



, J_ Jlixb)No{xP2) - Nl(xh)Mxp2) ^ MO j^^ P2 

 XP2 Jl(xP2)Ni(xh) - NyUp2)Ji(xb) fi °^Pl 



(364) 



If xi is the smallest positive root of equation (364), then the attenua- 

 tion and phase constants of the principal mode are 



Xi 



(365) 



2 V/i/e -g ' 



/3 = wV^ . (366) 



These expressions for a and /3 are of exactly the same form as equations 

 (295) and (296) for a complete Clogston 2, except that x\ is now de- 

 termined from equation (364) instead of equation (293). It is easy to 

 show that (364) reduces to (293) if there is no main dielectric, that is, 

 if pi = P2 . 



For any given values of the four ratios a/6, pi/h, po/b, and mo/m, equa- 

 tion (364) may be solved for xib by numerical or graphical methods. 

 However if we wish to examine many cases, so as to investigate the 

 effects of varying some or all of the parameters, a more efficient proce- 

 dure for finding xi is needed. Such a method is provided by the observa- 

 tion that in spite of the complicated appearance of equation (364), it is 

 really just the equation which determines the eigenvalues in a rather 

 simple two-point boundary- value problem, which is well adapted to 

 solution on a differential analyzer. We digress briefly to formulate this 

 problem. 



The differential equations for the fields in the main dielectric can be 

 put in the form of equations (67) of Section III, namely 



