CONSTANT PHASE DIFFERENCE 103 



of determining the roots of the polynomial A -{- pB . In the case of the 

 Tchebycheff type of characteristic described in the first part of the paper, 

 the required roots can be determined by means of special relationships. 



In the first place, the roots oi A -]- pB are the roots of ( 1 + i tan ;^ ) • In 



other words, by equation (4) they are the roots of 1 -f- iU dni nu— , kij \. 



The values of u at the roots turn out to have an imaginary part iK', where 

 K' is the complete elliptic integral of modulus \/l — k^. If a new variable 

 u' is defined by 



(17) u= u' + iK' 



the roots can be shown to correspond to the values of u' determined by 



cn\nu -^, ki j 



If it is assumed that the phase variation is small in the range of approx- 

 imation to a constant, it can be shown that one value of u' determined 

 by the above relation is given approximately by 



(19) ^ = -/3. 



where I3a is the average phase difference for the range of approximation as 

 before (in radians). After this value of u' has been computed, all the roots 



.,[ 



\ -\- iU dnxnu 



[nu— , kij 



can be found by computing the values of to 



hj 



corresponding to this value of u' and to those values obtained by adding 



2K / K \ 



integral multiples of the real period — of dni nu — ^ , ^i J. This gives the 



following formula for the roots in terms oi p = iw. 



(2aK . . 

 en I h Mo 



(20) ^'^'"'{ii ^' <r = Q,---,{n-\) 



sn I + Mo 



in which «o is the value of u' determined by (19). 



Finally, instead of using the above elliptic function formula directly, one 

 may replace the elliptic functions by equivalent ratios of Fourier series 

 expansions of 6 functions. This gives 



,o.x . / cos (X,,) + q^ cos (3XJ + if cos (5XJ • • • 



(,21j pc = Vcoia)2 -^ — TT-y- r—- — /o. N I 1—. — tfTn 



sm (Xa) — q^ sm (3X») + q^ sm (5X,) • • • 



