ELECTRIC WAVI'.-EI ITERS 309 



that when dissipation is neglected there is infinite attenuation at some 

 frequency within each branch of Xik. Formula (8), when Vk — 0, 

 gives in the attenuating bands where C7/. ^ — 1 



cosh Au{g) = 



1 + ^^'^'■• 



1 + (1 -ewu 



(17) 



in which g = ;;/, uim' , mm' m" , etc., for the .^/-types and higher orders. 

 The critical frequencies occur where the attenuation constant becomes 

 zero, i.e., at Vk = — 1, while the frequencies of infinite attenuation 

 occur where it becomes infinite at Vk = — 1/(1 — g^)- Since, when 

 Vk^O, (si/,/2i?)- = Vk, we have the following results: 

 At critical frequencies /o, /i, etc., 



zu- = ± i2R. (18) 



At frequencies of Infinite attenuation, /oco, /loo, etc., 



^u-=±-y££=, (19) 



^l - r 



the number of such frequencies being equal to the number of critical 

 frequencies. 



A very simple relation has been found between these two sets of 

 frequencies in the case of any multiple band pass :l/-type or higher 

 order wave-filter. Such a relation is given here for each of the four 

 general groups into which all classes of band pass wave-filters may be 

 divided, each group having n internal bands with or without low pass 

 and high pass bands. 



Group 1. — Low-and-» Band Pass. 



f Ox fix ■ • * f2nx = , . ,/o/l ' ' ■ /in- (-^) 



VI - r 



Group 2. — n Band-and-High Pass. 



fl^hx • • ■ /(2n+l)oo = Vl - r/1/2 • • • /^n+l. (21) 



Group J. — Low-7/ Band-and-High Pass. 



/Ooo/loo • • ■ /(2n+l)oo = /o/l * ' * fin+l- (22) 



Group 4. — n Band Pass. 



/loo/.. • • • hnx = /1/2 • • • /.>„. {li) 



For this group there is a further relation but it applies to the 



