276 THE BELL SYSTEM TECHNICAL JOURNAL, MARCH 1952 



Let [Lt] be the concrete linear correspondence defined by the list 

 \Lt]{p) which consists for each complex p of all pairs [[a], [b]] where 

 [aleVr , [6]eKr . 



The correspondence described by [Lt] is what Cauer defines as an 

 ideal transformer. He shows, loc. cit., how it can be defined as the 

 limiting case of a physical transformer. 



There is also a dual kind of device, described by a correspondence 

 admitting all [6]eKi for which 



^ = ^- = . . . = ^ 



Xl X2 Xm 



and all [a]eVi for which 



Xiai + • • • + X,„a,„ = 0. 



This also is an ideal transformer obtainable as a limiting case of a 

 physical one. 



19.21 The correspondence Lt is PR. 



Proof: We observe that Vt = (K^)", for let [aleVr , [^jeKr , and let 

 t be the common value of the Ur/pr ■ Then 



(a, 6) = Sa,6, = ^2p,6, = ^(2pA) = 0. 



The postulates are now all easily proved. We omit the details. 



19.3 Let V and K be 5-dimensional spaces. We interpret the 6-tuples 

 [v] and [A:] of 19.04 as representing vectors yeV, keK in a real frame. 



Let L be the correspondence between V and K formed by juxtaposing 



(i) the correspondence described by [Zdip)] relating components with 

 indices in the range n + 1 to d, 



(ii) the several correspondences described by ideal transformers, 

 relating components with indices in the ranges 1 to n and rf + 1 to b. 



L is PR because it is the juxtaposition of PR correspondences. 



19.31 Let U and J be s — n-dimensional spaces. We interpret the [(/] 

 and [j] of 19.04 as representing luV, je] in a real frame. 



19.32 Let C be the operation from J to K whose matrix in oiu' chosen 

 frames is [C]. Then C* operates from V to U with the matrix 



[C]* = [Cy. By these definitions, C is real. Let M be the correspond- 

 ence between U and J obtained by restricting L with C. Then there is a 

 frequency domain T m such that M is PR (18.3). 



19.4 By 19.09, [M] in our chosen frame is the correspondence estab- 

 lished between mesh currents and mesh voltages by the network of the 



