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BELL SYSTEM TECHNICAL JOURNAL 



tion, in studying actual systems which necessarily involve at least a 

 small amount of dissipation. 



The mates of positive and negative integral powers of p, including 

 the zero power, cannot be derived directly and definitely from the 

 Fourier integral (101) without the specification of an additional 

 passage to a limit. Such pairs therefore differ essentially from the 

 great body of regular pairs where the choice of one coefificient com- 

 pletely determines the mate. In order to permanently ear-mark these 

 limiting pairs, their serial numbers in Table I bear a star. These pairs 

 may be thought of as lying on the periphery of the great domain which 

 includes the totality of regular pairs. 



Identical Mates and Other Simply Related Mates 

 Since one of the coefficients of a pair may be assigned quite arbi- 

 trarily, this choice allows us, if we so elect, to specify some relation 

 between the two coefhcients of a pair. We might specify that a linear 

 combination \Fj{x) -f ixGj{x) of the two coefficients of a pair both 

 taken with the parameter x is to equal an arbitrary function F{x). 

 The pair {Fj, Gj) is then uniquely determined, unless X + i^'n = 0, 

 being equal to pair (224) after each Fn has been divided by X -f f"/x. 

 Again if it is specified that one coefficient is to be the reciprocal of the 

 other, a possible solution is pair (760). 



Fig. 5 — Identical coefficient pairs of the form 

 (1 + xyp')-hKi{2wpUl + xip')/Ki{2^p^-), X = / or g. 



The condition that the mates shall be identically the same function 

 of their parametric variables /and g is of special interest. In addition 

 to the identical pairs shown on Fig. 2, « = 0, 4, 8, the table contains 

 a number of identical pairs including (523), (625), (712), (761), (916). 



