268 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



of voltages yeVi such that [v, 0]€Li(p) for all peFi . Dual definitions 

 for Km, K a/0 , and symmetrical ones for Vat , • • • , Katq need not be 

 repeated. 



It is clear from these definitions that the four manifolds above are, 

 in the order listed, the manifolds 



Vl , Vlo , Kl , Klo 



for L. Now, for example, 



(Kz,o)° = (Kmo e K^o)" = (K,,o)»n(K^o)'' 



by 10.6. This last manifold, in V, is (V„ V.) (1 (V^ Vi), byP3 

 for M and A^, and by 10.6. But by direct calculation 



(Vm V2)n(v^ Vi) = Vm v^ = v^ . 



The dual of this result then completes P3 for L. 



P4 for L is immediate because the Ei and Ei are real. 



The duality of the decompositions of V and K implies the identity 



(v, k) = {Eyv, Etk) + {E.yv, E*k) 



(that is E1E2 = E2E1 = 0, and dually. This is Halmos^, par. 33). All 

 of P5, P6, and P7 for L follow at once from this identity. 



17.22 The "only if" of 17.2 is a special case of the result of Section 18. 

 Its proof will be deferred to 18.4. 



17.23 It is obvious that the notion of juxtaposition and the lemma of 



17.2 extend to juxtapositions of more than two correspondences. 



17.3 Even without the "only if" part of 17.2, we have enough for the 

 following characterization of PR correspondences: 



Theorem: A correspondence L is PR if and only if it is the juxtaposi- 

 tion of 



(i) a correspondence defined by a nonsingular PR matrix between 

 a Vi and a Ki = Vi , 



(ii) a correspondence consisting of short circuits: that is of pairs 

 [0, k] for all keK-y and all p, 



(iii) a correspondence consisting of open circuits: that is, of pairs 

 [v, 0] for all f eVs and all p. 



Proof: If L is PR, the decomposition indicated is that of 13.1, 13.11, 

 13.12. If L is the juxtaposition indicated, then it is PR by 16.6 and the 

 "if" in 17.1, provided the short and open circuits are PR correspondences. 

 The verification of the postulates for these latter is easy and will be 

 omitted. 



