LAMINATED TKANSMISSION LINES. II 1173 



rcctions will be the exti'cmie inner layers, wliicli occupy at most a small 

 traction of the total volume of the stack. 



The average current density Jz in a Clogston 2 with infinitesimally 

 thin laminae is given by equation (305) of Section VTTT; namel}^, writing 

 Xp for the pth mode and dropping e"'\ 



where Ifu is an arbitrary amplitude constant. For n — and 1, CnixpP) 

 denotes tli(> coml)ination of Bessel functions 



CnixpP) = N,(xJ>).f„(xpP) - Ji(xph)yn(xpP); (474) 



and X;- •■'^ tli(' /^th positive root of 



Ci(xa) = Ni(xh)J^(xa) - ./i(xM.V,(xa) = 0. (475) 



According to ecjuation (434) of Section X, we may wi'ite 



2)Trfpia/b) , . 



Xp = —, • , (4/()) 



b — a 



where the functions fp{a/h) are of the order of unity. The total current 

 /(p) flowing in the positive ^-direction between the inner core and a 

 cylinder of arbitrary radius p is just 



Up) = 2ir f pJ, dp = 27r//opCi(xpp). (477) 



•'a 



The thickness of the jth conducting layer in a stack of finite layers 

 may be written 



t, = e(k + ^2) = d(pj - p;_i) = dip, , (478) 



where Apy represents the thickness h + 1-2 of the jth. double layer. Hence 

 approximately 



II j = 2Trpj^J^Apj = 2TrHoXppj-iCi)(xppj~i)'^pj , (479) 



it being remembered that the conduction current in the conducting 

 layer is essentially equal to the total current in the double layer, since 

 the displacement currents are negligil)le. The cuircnt flowing inside the 

 radius py-i is, from (477), 



7y_i = 2irHnPj-.iCi(xpPj~i), (480) 



and so the {)ower dissip;itc(j pci' unit Icuglli in tlic./lli c(»iiductor is 



