566 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



analogous to (2) and (7), mth the scalar product 



n n 



JLerttr+Hfrhr. (9) 



r=l r=»l 



It is a common and convenient malpractice in vector algebra to use, 

 for example, the symbol j both for an m-tuple in J and for the ?i-tuple 



i e J e Ki 



of the form (8). Taking this advantage, we can see that (9) is simply 



(wi + vi , ii + A) + {U2 + v-2 , J2 4- ^2) (9') 



where here the parentheses denote scalar products between V and K. 

 The form (9') can also be derived directly from (1), (7), and (I, 10.6). 



4.55 We now wish to compute the voltage-current pairs admitted by 

 M.vD • Referring to Fig. 5, we observe that Nx and Nl both have im- 

 pedance matrices (X(p) and L(p) respectively, or, rather, the matrix 

 forms of these in the frame of present interest) finite at all p except 

 p = 0, p = 0° . Each will, therefore, admit any current n-tuple into its 

 terminals, i.e., through its ideal branches, at any but these exceptional 

 frequencies. By construction, Ng has a non-singular admittance matrix 

 and therefore also will admit any current m-tuple into its terminals 

 (2.07), except at most at certain isolated frequencies. It is evident by 

 Kirchoff 's laws applied to Fig. 6 then that M ad will admit an}^ current 

 2n-tuple of the form 



(ii e /:) © 0-2 © i-k)) (10) 



where ji e], i = 1,2, and fc e Ki , except at most at finitely many ex- 

 ceptional frequencies. Conversely, if the current 2/i-tuple specified by 

 (7) is that in Mad , conservation at the absent shunt arms of the lower 

 degenerate T-sections implies that, as elements of K, 



Ai + h = 0, 



that is, the current is of the form (10). Hence 2n-tuples of the form 

 (10) span the space of currents admitted by Mad • Let us call this space 

 Km . It is a proper subspace of K" unless ?n = n. 



4.56 Let G~^(p) denote the m X m impedance matrix of Ng . Then by 

 (7) of 4.4, interpreted as an operator equation, 



G-(p)=(i. + !)«-■ (H) 



where G~^ is a real, constant, symmetric, non-singular m X m matrix. 



