FORMAL UKAI.IZAUILITV THlOOltY 1 277 



2n-pole N. When this network operates as a 2/t-pole, the only mesh 

 voltages which are not zero are tiiose relating to the external meshes, 

 since there are no internal sources of x'ollage. We must now account 

 for this. 



19.41 Let Y-> , K2 be /i-dimensional spaces. Choose a real frame and let 

 1) be the operation which takes 



[ai, ■ ■ ■ , dnUV' (3) 



into 



[n,, ■■■ , «,. ,(),•••, OleU (4) 



in the frame of 19.31. Then /) is real and J)* in the chosen frames takes 



[bi , • • • , hhJ (o) 



into 



[hi, ■■■ , K]eK, . (6) 



19.42 We interpret the n-tuples (3) and (6) as voltages and currents 

 in the external meshes of N. Their relations to (4) and (5) are con- 

 sistent with this interpretation. 



Let us restrict M by D, to get a correspondence Mi between V2 and 

 Ko . In our chosen frame, the passage to [Mi] corresponds, by (3) and 

 (4) of 19.41, to considering mesh voltages in N which vanish for every 

 internal mesh, and, correspondingly letting the mesh currents adjust 

 themselves to this situation. We of course observe only the external 

 mesh currents (6). 



19.43 M was PR. So, therefore is il/i (18.3 dual). Since [Mi] is the 

 correspondence established by the physically realizable 2n-pole N, 

 the necessity of PI, • ■ • , P7 for formal realizability is established. 



XX. APPENDIX TO PART I 



20.0 We must prove 7.22 and those assertions of 10.0 which are not 

 covered in Halmos'. These concern reality. 



20.1 Let Vi be a real manifold and 



V = Vi e V2 , K = Ki e Ko 



where Ki = (V.)", etc. The basis (14) of lO.G exists by Halmos^ par. 19. 

 We show that it can be chosen to be real. We have linearity independent 

 vectors 



Vl , • • • , Vr , Vr + \ , • • • , Vn , 



