CONSTANT PHASE DIFFERENCE 99 



a "parametric variable" which would be eliminated on forming a single 

 equation from the two simultaneous equations indicated. The modulus ki, 



of the dn function corresponding to tan ( ;^ ) is related to the modulus k, 



of the dn function corresponding to co, in the manner indicated below. The 

 constant Ki, of course, represents the complete integral of modulus k-[, 

 just as K represents the complete integral of modulus k. 



Corresponding to any modulus k there is a so-called modular constant q. 

 Using ^1 to represent the corresponding modular constant of modulus ki, 

 it is here required that 



(5) qi = q\ 



One modulus can be computed from the other by means of this relation- 

 ship and tabulations of logio q vs sin~^ k which are included in most elliptic 

 function tables." 



Degree of Approximation to a Constant Phase Difference 



When M is real and varies from zero to infinity, the corresponding value 

 of CO as determined by (4) merely oscillates back and forth between the values 

 0)1 and C02. In other words, it merely crosses back and forth across the range 



in which tan ( - j approximates a constant. Similarly, when u is real and 



increases from zero to infinity, tan ( - j oscillates between U\/l — kj and 

 U. The equal ripple property of the curve illustrated in Fig. 4 is explained 

 by the fact that the period of oscillation of tan ( - j with respect to u is 



(9 



merely a fraction of that of co, so that tan ( - ) passes through several ripples 



while the value of co moves from coi to co2. 



Combining the formulae for the maximum and minimum values of 



tan l-j gives the relation 



(6) tan('^U^('-^/'"^9 



2/ 1 + UWl - kl 



^ When k is extremely close to unity, it may be easier to obtain accurate computations 

 by using the additional relation 



logio iq) logic iq') 



\!oge (10;/ 



OJl 



where q' is the modular constant of modulus y/l — k^ 



