648 BELL SYSTEM TECHNICAL JOURNAL 



(3) Any linear combination of pairs is also a pair. Cf. pairs (201), 

 (204). 



(4) The odd and even parts of every pair are also pairs. 



(5) If both coefficients of a pair are real, both are even. 



(6) If a pair has one real and one pure imaginary coefficient, both 

 are odd. 



(7) If a coefficient is even and real, its mate is also even and real. 



(8) If a coefficient is odd and real, its mate is odd and pure imagi- 

 nary, and vice versa. 



(9) If a coefficient is real, its mate has conjugate values for opposite 

 values of its parameter and conversely. Cf. pair (216). 



(10) The conjugates of the coefficients of a pair are also a pair pro- 

 vided the sign of either frequency / or epoch g is reversed. Cf. pair 

 (215). 



(11) A pair with the signs of both frequency/ and epoch g reversed 

 is also a pair. Cf. pair (214). 



(12) The parameter of either coefficient may be multiplied by a 

 positive real constant provided the other parameter and coefficient are 

 each divided by the same constant. Cf. pair (205). 



(13) Coefficients of a pair may be interchanged if, when interchang- 

 ing the parameters, the sign of one parameter, either /or g, is reversed. 

 Cf. pair (217). 



(14) Any pair may be resolved uniquely into the sum of four pairs 

 by pairing together: the even, real parts; the even, imaginary parts; 

 the odd, real part of each coefficient with the odd, imaginary part of 

 the other coefficient. 



(15) A pair may have the form (F(f), XF(g)) where the multiplier 

 X is constant, if and only if X has one of the four unit values (1, i, 

 — 1, — i). Such a pair is called an ^"-multiple pair. Cf. pair (223). 



(16) Any ^'"-multiple pair has both coefficients odd or even according 

 as n is odd or even. 



(17) Any ^"-multiple pair with complex coefficients may be resolved 

 into two ^"-multiple pairs with coefficients which are real or pure 

 imaginary. 



(18) The coefficients of any two ^"-multiple pairs are orthogonal if 

 the {"■ multipliers are different. 



(19) The coefficients of any four ^"-multiple pairs with different i" 

 multipliers are linearly independent. 



(20) Any pair may be resolved uniquely into the sum of four i"- 

 multiple pairs; i.e., pairs of the form F„(/), j"F„(g). Cf. pair (224). 



(21) Any pair may be resolved uniquely into the sum of eight i"- 

 multiple pairs where Fn(f) is real or pure imaginary. Cf. pair (225) 



