568 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1952 



In the present special circumstances it is almost as easy to study Mad 

 in a slightly different way than this. Since fewer direct references to I 

 are involved, we shall take the alternative path. 



We first calculate the scalar product between the voltage (14), (15) 

 and an arbitrary current of the form (10), say the current 



Qh ® () @ (ho ® (-{)) eKli. 



To do so, we consider the form (9') for such a product. In the first writing, 

 then, this scalar product is 



(XipXJx + k) + G-\p)(h + h) + vo , h + () 



+ {L(p)(j, - k) + G~\p)(j, + j,) + Vo , h, - ^). 



Each of these scalar products has three voltages appearing in it. Dis- 

 tributing the products over these voltages, and using the facts that the 

 range of G~^(p) is J and that I'd e Vi = (J)° we get a second form: 



(X(p)(ii + /.■), Ih + + {G-\p)Ui + J2), h,) + (%, () 



+ (L(p)(j, - k), h, - i) + iG^\pM + Jd, Ih) + (.0, - ^). 



The terms involving Vo go out and we can collect to 



(X(p)(:/\ + fc), ih + () + {G-\p){j, + i2), h, + /!,) 



+ (L(p)(j, - fc), h, - I). 

 This is the desired scalar product. 

 4.58 Let us now consider the {n + m) -tuples 



[oi , 02 , • • • , a„ , &i , • • • , hr,] = ji @ k @ J2 (17) 



obtained from (2) by deleting the hm+x , • • • , b« . We still interpret these 

 as currents into the relevant terminals of Mad • We also observe that 

 when the current (17) is given, (2) can be determined, because by (10) 



a„,+s + hm+s = 0, s = 1, 2, • • • , n - w. 



Given (17), and therefore (2) or (10), we can determine the voltages 

 (14) and (15), where Vo is an arbitrary element of Vi . Let us agree now 

 always so to choose Vo that the component of (15) in the subspace Vi 

 vanishes. This means that, in (17), we have specified arbitrarily the cur- 

 rents into the left-hand terminals of M ad (on A-A) and into the upper m 

 of the right-hand terminals. We have also agreed that the voltages 

 across the lower n-m terminals on D-D shall be zero, so that (15) is an 



