62 BELL SYSTEM TECHNICAL JOURNAL 



of the curves in this cycle may be used to obtain the values oi 1 — K 

 and i/' for any value of ^2 for which the curves go below the edge of the 

 charts, as follows. Multiply ^2 by the smallest power of ten that will 

 give a value on the chart, read the value oi I — K or ip corresponding to 

 this new ^2 and divide this value of 1 — -K" or 1/' by this power of ten to 

 obtain the desired value of 1 — iiC or ^. To obtain values oi I — K and 1/' 

 for larger values oi q = 2(r// than shown on Figs. 5-12 use is made of 

 the fact that both of these quantities may be expressed as functions 

 of the parameter V5 sin ^2 for large values of q, (Q « 1). That is, the 

 shape of all the curves for values of Q that are small compared with 

 unity is the same. Hence to obtain values oi 1 — K and ^ for values of q 

 greater than those for which curves are shown, divide the given q by 

 some power of one hundred that gives a value of q for which a curve is 

 drawn, and read the desired value of I — K or xj/ opposite the value of 

 sin ^2 that is the same power of ten times the given sin ^2 as the power 

 of one hundred by which q was divided. 



If the following characteristics of the curves are taken into con- 

 sideration, interpolation is simplified. The similarity of the i^T-curves 

 suggests relabelling the abscissa so that the value of K for some 

 intermediate value of q may be read from one of the curves that is 

 drawn. Any curve for a large value of q, {Q <K 1), such as for q = 10" 

 may be relabelled q = x X 10" if the value of the abscissa is divided by 

 V^. The same method is useful for small values of q but in this case 

 the quantity by which the abscissa must be divided depends on the 

 value of Q. Also in this case the shape of the curves changes with Q 

 so that it is desirable to read the values from the curves drawn for the 

 nearest values of q on either side of the desired value. The factor by 

 which the abscissa must be multiplied to obtain the desired result may 

 be inferred from the interpolation scale on the curves. When q is 

 large the same method of interpolation may be employed for the 

 ^-curves as for the jK'-curves. When q is small, (Qy^ 1), the fact that 

 \l/ is proportional to q suggests the method of interpolation. (On 

 vertical polarization when ^2 is greater than the Brewster angle, -k — \p 

 is proportional to q.) If the value of 4' for 5 = .r X 10""" is required, 

 read the value of 1^ from the curve for q — 10~" and multiply by x to 

 obtain the desired value of yp. If greater accuracy is required the values 

 may be calculated from the equations and Table III of Appendix II 

 without prohibitive labor. 



When the antennas approach the ground, the ratio given in equa- 

 tion (21) approaches zero so that more terms of the complete solution 

 must be taken into consideration. Wise ^^ has derived an expression 

 for the effect of the ground on the Hertzian potential which when 



