ELECTRIC WAVE-FILTERS 333 



When r = <x , meaning anti-resonance of z, each anti-resonant frequency 

 appears twice in the product. 



Form 4. F^n = — , independent of c. 



fl2n 



When c = 0, meaniyig resonance of z, each resonant frequency appears 

 twice in the product. 



To prove the theorem for Form 1 first square the expression in (51) 

 and clear the fraction. This gives a polynomial in /^ of degree 

 2w + 1, of which only the terms of highest and zero powers need be 

 shown for our purpose. Thus 



(j2)2„+i _,_ -1^ = 0, (55) 



which expresses the general relationship between x^ and /-. If x^ is 

 given some constant value as x^ = c^, that is x = ± c, the roots of (55) 

 will be the 2w + 1 distinct values of /^ where x = ± c. By the theory 

 of equations, the product of these 2n -\- I values of /^ is (c^/aon+i). 

 Since we are interested only in positive frequencies, we may take the 

 positive square root of both sides with the result that the product of all 

 frequencies at which x = ± c is c/a2n+u which proves the theorem. 



The proofs of the theorems for Forms 2, 3 and 4 are exactly similar 

 and should not need further explanation. In Form 3 the values 

 .X- = -£- GO occur at the anti-resonant frequencies of s, namely fia, f^a, 

 etc. ; hence, when c = oo the total frequency product includes each of 

 the latter frequencies twice. The result for Form 4 has a meaning 

 even at the limit c = 0. These frequencies are the resonant ones of s, 

 where z = 0, and each one of them must obviously appear twice in the 

 total product. 



Proofs of Wave- Filter Frequency Relations 



As was stated in Section 1.9, Zik satisfies certain conditions at tiie 

 particular frequencies of interest. 

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



zik = ± i2R. (56) 



At frequencies of infinite attenuation, /o^), /ico. etc., 



zik = =L (o7) 



V 1 - 2- 



