ELECTRIC WAVE-FILTERS 287 



5 will also have identical transfer constants which are functions of s. 

 At the limiting value of the parameter, 5=1, each derived structure 

 becomes identical with its prototype. The reason for the use of 5 as 

 the general parameter instead of m, as in previous papers, is to permit 

 it to take on without confusion a succession of values including m, as 

 will be seen. 



Beginning with the "constant fe" wave-filter of any class as the 

 initial prototype, these two operations are performed alternately on 

 successive structures, which results in producing two different sequences 

 of wave-filter structures, depending upon which of the operations is 

 first used. These wave-filters are all of the same class and contain 

 successively more and more elements. In Sequence 1 (see Fig. 4) the 

 first operation is Pi(w), then D-iim'), Di{m"), etc., the parameters being 

 taken in succession as 5 = w, m' , m" , etc. In Sequence 2 (see Fig. 5) 

 the first operation is Di{m), then Di{m'), D^im"), etc., with the same 

 succession of parameters as before. Since at each derivation another 

 single parameter is introduced, each successive structure of either 

 sequence has one more arbitrary parameter than the preceding struc- 

 ture and the number of arbitrary parameters in any structure is equal 

 to the number of alternate operations performed to obtain it from the 

 ' ' constant k ' ' wave-filter. Now every section has one mid-point image 

 impedance which is a function of all of its arbitrary parameters. Hence, 

 this whole process is effectively one for obtaining a structure with an 

 image impedance which contains any desired number of arbitrary 

 parameters. The first derived structures in both sequences are the 

 pair called M-types having the parameter m. The second derived 

 ones will be called the pair of il/M'-types with parameters m and m' ; 

 the third, the pair of J/.l/M/"-types with 7n, m' and m" \ etc. Each 

 successive pair can have a more nearly constant resistance impedance in 

 all transmitting bands than the preceding pair because of one additional 

 parameter in the image impedance functions. The two members of a 

 pair have identical transfer constants and either member can be 

 obtained from the other, as inverse networks of impedance product B?. 



While the derived structures are wave-filters having the same 

 transmitting bands as the "constant ^" wave-filter, their propagation 

 characteristics are otherwise more general. However, no different 

 propagation characteristics are obtained in the successively derived 

 structures than are possible with the first derived or M-types since 

 all these derived structures have potentially identical transfer con- 

 stants, the transfer constant of any structure being dependent upon 

 its parameters only in their product. A simple relation is given 

 here between these parameters, the frequencies of infinite attenuation 



