594 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



-mt + (JPhP - PHVH)Ht= V^. , 



which yield 



{[veVh{^ + PePh) — iS ] — JvhVe{ph + Pe)P]Ei 



= -i/3V^. - v„{jP - PhWH,, 

 and 



{[V£j'fl(l + p^p//) — iS"] — JvhVe{ph + PE)P]Ht 



The term in parentheses may be removed by using the rule for inverting 

 such expressions in P which was given earlier. This process gives 



i2 



^Et = 



8 



1 + PePh — ) + j{pH + Pe)P 



VeVh, 



(20a) 



and 



mt = 



where 

 12 = veVh 



= VeVh 



1 + PePh — 



VeVh 



[- i|3VE. - VhUP - Ph)VH.], 



+ jipH + Pe)P 



(20b) 



MjP - pm)VE, - i^VJYJ, 



1 + PePh 



- I\ 

 VeVh/ 



(pe + PhY 



_VeVh 



- (1 + Pe) (1 + Ph) 



&- 



VeVh 



— (1 — p£)(l — Ph) 



It may be noted that for plane waves in the unbounded medium along 

 the z axis, which have Ez = H^ = 0,0, must vanish and that the propaga- 

 tion constants for such plane waves are evidently given by 



I3~ = VEVnil ± Pe) (1 ± Ph)- 



(21) 



The values of E^ and Hz given by (18) may now be substituted in (20) 

 and the operator P removed. This gives, finally, 



(Ai — A2)UEt = j ( 1 + PePh — ) il^M — PhVh) + Vh{ph + 



L\ VeVh/ 



Pe 



[ 



— (i8A2 — PhVh)(pe -\- Ph) + Vn [l + PePh 



VeVh 



(22a) 



v*h 



minus the same expression with suffixes 1 and 2 interchanged. 



