88 BELL SYSTEM TECHNICAL JOURNAL 



Now let us return to formula (229), and expand in inverse powers of 

 p : we get 



A _ 1 I 1 2» + 2 1 I (2n+3) ( 2n+4) 1 ) 



22Vp^'p'' 2^1! p^+i"^ 2*2! p«+2 ••••f*^^'^^^ 



Now since \/p2i02 = ~ir, we have 



__2_j/ 2 \" 2w+2 / 2 \"+i 



(2??+3)(2^+4) /• 2 \ «+2 _ 

 "^ 222! Vi?C^y 



Replacing 1/^" by t"/n\ we get finally 



2 [ 1 /2^ y ( 2/Z+2) /2/ \"+^ 

 ''~2"i? [w!V]?cJ 2.1!(w+l)!\i?Cy 



(27z+3)(2w+4) /2^y'+2_ 

 ■^ 22.2!(w+2)! \RC) 



(238) 



For large values of n and ^ this series is difficult to compute or in- 

 terpret. It can, however, be recognized as the series expansion of the 

 function 



where I„{2i/RC) is the Bessel function /„ of order n and argument 

 {2t/RC). This solution, it may be remarked, can be derived directly 

 by a modification of the integral formula (w). 



It is beyond the scope of this paper to consider other types of 

 artificial lines and wave filters; for a fairly extensive discussion the 

 reader is referred to "Transient Oscillations in Electric Wave- Filters," 

 B. S. T. J., July, 1923. The low-pass wave filter, however, both in 

 its own right and on account of its close relation to the periodically 

 loaded line, deserves further discussion. 



For the non-dissipative low-pass wave filter, we have 



A^„=l£'''^JUr)dT (234) 



while for the quasi-distortionless low-pass wave filter 



^«=^J^'^'^e-^V2„(T)JT (235) 



where /x = X/coc = R/Lcoc = R/2vL. 



