98 BELL SYSTEM TECHNICAL JOURNAL 



In these equations, the symbol sn indicates an elHptic sine, of modulus 

 k, while A' represents the corresponding complete elliptic integral. U is 

 merely a constant scale factor, while n is an integer measuring the complex- 

 ity of corresponding networks. In the case of phase-difference networks, 

 n represents the total number of sections of the type indicated in Fig. 3, 

 which are included in the two phase-shifting networks or their tandem sec- 

 tion equivalents. 



In Cauer's filter theory, the variable X represents a rational function of 



CO which permits F to be an image impedance or a coth ( - ] function. In 

 order that F may be an odd rational function of oj, however, as is required 

 when it is to represent tan ( - j , X must be defined by the relation 



(2) (0 = C02V1 - X\ 



Cauer shows that F approximates a constant in the Tchebycheff sense in 

 the range < X < ^ . Hence, in terms of o, the range of approximation 

 is coi < CO < C02 , where coi and 002 are arbitrary provided the modulus k is 

 assumed to be determined by the relation 



Vol - 



(3) k = ^ "^ ~ "■ . 



Alternative Expression for the tan ( - j Function 



While equations (1) are the most convenient form of F to use in deriv- 

 ing the transformation of the variable, an alternative more compact form 

 is more suitable for determining the degree of approximation to a constant 

 phase difference and the element values of corresponding networks. When 



F represents tan I - j and hence co and X are related as in (2), the equivalent 



expression is as follows:^ 



tan I - 1 = Udnxnu-— 

 (4) \2/ \ A 



CO = C02 dn{u, k). 



In this expression, dn represents a so-called "</«" function, the third type of 

 Jacobian elliptic function usually associated with the elliptic sine, or sn 

 function, and the elliptic cosine, or en function. The symbol ii represents 



^ This expression depends on a so called modular transformation of elliptic functions 

 not found in the usual elliptic function text. The transformation theory may be found in 

 "An Elementary Treatise on Elliptic Functions," Arthur Cayley, G. Bell & Sons, Lon- 

 don, 1895. 



