334 BELL SYSTEM TJiCHNICAL JOURNAL 



l-'very negative or positive branch of Zu: includes one each of these 

 frequencies. 



For those wave-filters with only internal transmitting bands the 

 additional relation will be used which specifies the frequencies where all 

 image impedances equal R and the series impedances become resonant. 

 At these resonant frequencies, fir, fir, etc., in the transmitting bands 



zik = 0. (58) 



We know that in a "constant ^" wave-filter the transmitting bands 

 include the frequencies at which the series impedance Su is resonant. 

 Hence, to the four forms of impedance function for Ziu, as in (51) to (54), 

 there correspond four groups of wave-filter classes as already men- 

 tioned. These groups were designated according to the general 

 locations of their transmitting bands which obviously correspond to the 

 locations of the resonant frequencies of su-. For this reason each wave- 

 filter group and the corresponding impedance form of Su have the same 

 number designation. 



Group 1. Low-and-w Band Pass. 



An application of the theorem for Form 1 with (56) and (57) gives 

 immediately the desired relation (20) 



./Ooo/loo • • • /2nco = f ^ „ -•/"./> ' ■ * /2»- 



VI - g" 



Similarly the relations (21), (22) and (23) are obtained for Groups 2, 

 3 and 4. Relation (24) for Group 4 is derived from (56) and (58), the 

 latter corresponding to c = in the theorem for Form 4 where each 

 resonant frequency appears twice; the square root of the resulting 

 relation is (24). 



