FORMAL REALIZABILITV rilEORY — II 573 



Now let V be an arbitrary voltage in V and let 



lu = Z(0)Ev. 

 Then w e V, and by (9) the current 



Ew = 

 for any v. Hence 



= (v, Ew) = (w, E*v) = (w, E*v) 

 = {Z{0)Ev, E'v) 



(11) 



by (I, 7.2, 14.0). Now E is real and symmetric, as noted above. Hence 

 E ^ E — E' . Furthermore, Z(0) is real, so (11) becomes 



{Z{Q)Eu, Eu) = (12) 



where m = v is an.y element of V. Now Z(p) is non-singular on p = loj, 

 and its real part is semidefinite there. At p = 0, Z(0) is its own real 

 part, hence semidefinite and non-singular, hence definite. Then (12) 

 implies that En = 0. This being true for all u e VJE = 0. 

 The proof that Q = follows similarly from (10). 



4.G4 With Y^' (p) and F^^^(p) simplified at p = and oo^ we can go back 

 and compute 



1 = Z"\p)Y'"(p) 



= (z(p) + iA + ,ij)(^G+r»(p)). ^''^ 



Of the six terms obtained on expanding this exactly one, namely 



^-AY''\p) 

 V 



is not ob^•iously finite at p = 0, and another, 



vBY''\v) 



is not a priori finite at p = oo . We conclude by multiplying through by 

 p and letting p ^ 0, and dually at p = oo ^ that 



AY^'\0) = = F'''(0)A 



(14) 

 BY^'\o.) = = y^'^(oo)/?, 



where the commuted form can be established bj^ a new calculation from 

 1 = Y'(p)Z~(p), or by taking transposes. 



