316 BELL SYSTEM TECHNICAL JOURNAL 



where the summations extend as above from 1 to n. 



Writing the equation around the loop used in deriving equation (15) 

 it is found that: 



Z(Zi(« + Z2W - 2Zi2(*))4:13 



= E(Zl(« - Zi2(«) - Zi3a2 + Zi3:34 (22) 



= i:(Zl(« - Zi2(^>) - KZl2:12 + Z34:34 " 2Zi2:34), 



the last step being made by use of the reciprocity theorem and the 

 formulas already developed. Thus, finally: 



Zi3:l3 = Z + J'^(Zi2;i2 + Z34:34 " 2Zi2:34), (23) 



where 



r, ^ ^ ZlWZ2^^) - (Zl2(^^)'^ 



.=iZi(^)+Z2(« -2Zi2^*>* 



As already mentioned, Z is the impedance between short-circuited 

 terminals 1, 2 and 3, 4; this may be verified in a number of ways. 



The remaining impedances follow by superposition, which can be 

 carried out formally through the impedance notation in the manner 

 suggested. There are 21 driving-point and transfer impedances 

 between terminals which can be displayed in a triangular array similar 

 to that shown on Fig. 2B. The additional measurable impedances at 

 the terminals arise as follows: 36 from short-circuiting two terminals, 4 

 from short-circuiting three terminals and 3 from short-circuiting 

 terminals in pairs. 



Equation (22) may also be written in terms of currents ik-.n and 

 4:34, since Zi3:i2 and Zi3:34 may be expressed in terms of the latter; this 

 suggests the following relation: 



4:13 = V + vik:\2 — vik:5i. (24) 



The relation is verified by substituting into the mesh equations for 

 currents 4:i3; the typical equation is as follows: 



- Z(*-1>4-1:13 + [Zi(*) -f Z2<*) - 2Zi2<'=) + Z(^-" + Z<«]4:13 



- Z«4+l:13 = Zi(« - Zi2<«. 



The remaining current relations then follow by superposition, as 

 follows: 



