i.o.inr.n uxf.s .ixd comfens.itixc xr.riroRKs 4.w 



throuKluml tlu- first Iraiisinitiing luiiul (()<r<l) aiul a CDiisitliT.iMe 

 part of the surri'i'ilini; altrnuatiii^ liaiul; hut depart widely beyond. 



The exact forimilas for Z. Il'.ind /.'. If" for any lerminalions a and a' 

 (an he written in the forms 



Z=l ^ ^.» cot (2g- l)D+Vx7(iTx) (27) 



W' cot (2a-l)D+iZ.6V(l+X)/X ' 



Z' = l.^Z'. + i^^^. (28) 



" \/X(l + X) 



These equations are not restricted to values of a and c' less than 

 unity; they are valid for any (real) values of these quantities. When 

 \ = 0, they reduce immediately to (4) and (5) respectively. 



From (27) and (28) it is readily verified that Z and Z' are pure 

 imaginary throughout ever>' attenuating band, and it can be easily 

 seen that they are complex througliout every transmitting band; 

 because Z 5 and Z'j are pure iniaginar\- throughout every attenuating 

 band, and pure real throughout every transmitting band. 



It is seen from (27) and (28) that, throughout every transmitting 

 band, each of the quantities Z, IV, Z', W changes merely to its con- 

 jugate when <r is changed to l—a. Thus the conjugate property 

 expressed by equations (8) is not limited to loaded lines without 

 distributed inductance but holds when there is any amount of dis- 

 tributed inductance. Thus it continues to be true that complementary 

 characteristic impedances are mutually conjugate — throughout every 

 transmitting band. For Z' and W, these facts are readily seen from 

 physcial considerations also; though not so readily for Z and W. 



From physical considerations, as w'ell as from equation (28), it is 

 readily seen that Z' continues to possess the property expressed by 

 the second of equations (9); on the other hand, IF no longer possesses 

 the property expressed by the first of (9). 



We shall now return to the important formulas (2.5) and (26) for 

 the mid-point relative impedances in order to discuss them for small 

 values of X such as occur in practice, and particularly for a frequency- 

 range not greatly exceeding that of the first transmitting band. For 

 this purpose it is advantageous to write these formulas in the following 

 forms, notwithstanding some sacrifice of compactness: 



1-, ?^^"^ -. (30) 



