LOADED I.I NFS AND COMl'liXSAIIXG XF.rilORKS 457 



soniiU-nt. Thus, if />, (ienntt-s tlu- /ito of sinli-T situ.iti'd in llic ;;ih 

 segment, then 



(«-l)^ <£»,<« 2- (12-A) 



I'.itht-r .malytically or RraphicallN' it is rcadil\' seen that, wiien \ is 

 Muall, Dn is only slightly greater than (« — l)jr 2; it approaches that 

 \alue as a limit when « approaches infinity, for all finite values of X. 

 The [xnver series formula (21) for D„ is derived at a little later point 

 in this Appendix. 



Formulated anahtically, with the arguments of the trigonometric 

 functions reduced to the smallest positive values that preserve the 

 values of the functions, the transition values of D are the values of 

 /?,.,+! and Z), satisfying the equations 



sin=2(z?,.„+,-«^j =0. (1.3-A) 



/;„tan(/)„-[//-l]y)=X. (14- A) 



with «=0. 1. 2. :i. . . . in (13-A) and m = 1. 2, 3, . . . in (14-A). Equa- 

 tion (13-A) is equivalent to sin-2Z? = 0. With n odd and with n even, 

 (14-A) is equivalent respectively toDtanZ) — X = Oand to DcotD-\-\ = 0. 

 An equivalent of (14-A) is obtainable from the second factor of (3-A). 

 By (3. 1-A), still another equivalent is Z' 5 = 0; that is, the values of Z)„ 

 are the zeros of the mid-load relatixe impedance Z' .,. and hence of 

 the mid-load impedance A" >. 



With (n — l)ir 2 denoted by d„. equation (14-A) shows that 



D,-rf,<X rf„ (« = 2, 3. 4. . . .) Di<\/>^- 



By insfiection of (2-A) it can be readily verified that sinh-F is 

 negative when D^-i„<D<Dn and positive when Z)„<£' <Z>„ „+i; 

 and hence that these two ranges of D are a transmitting band and an 

 attenuating band, respectively, the corresponding compound band 

 thus being the range D„_\ „<D <D„„j^^. In this connection it 

 may be of some academic interest to note that, strictly speaking, 

 £) = is not a transition value of D between a transmitting and an 

 attentuating band. For (2-A) shows that sinh-F does not change 

 sign when D passes through 0; on the contrary, sinh-F is entirely 

 unchanged when D is changed to —D. Thus, D=0 is a point of 

 symmetry, but not a transition point. 



The values of £>,, namely, the roots of (14-A), cannot be written 

 down directly or expressed exactly. But they can be found to any 



