560 BELL SYSTEM TECHNICAL JOURNAL 



the same places in sets having the same impedance as the h'nes. 

 Transmission upon either circuit can then give rise to no crosstalk 

 upon the other circuit. Circuits such as two well transposed pairs on 

 a pole lead are here to be understood; that is, there may be any number 

 of circuits in the system but the mutual impedances and admittances 

 between conductors connect points which are equi-spaced with respect 

 to the impedances in each conductor. 



Now suppose that at a point distanced x from the transmitting end 

 of the circuits, slight alterations are made in the impedances, mutual 

 impedances and admittances of the system. The effect of each change 

 will be small and the total effect will be approximately equal to the sum 

 of the individual results. That is, we may neglect the second-order 

 terms, or crosstalk of crosstalk. Furthermore, unbalancing one of 

 the given circuits alone cannot produce crosstalk. It is necessary that 

 both circuits be unbalanced simultaneously by a single change in the 

 system. Now, adding impedances to either side of one of the given 

 circuits or to any third circuit will not unbalance both of the original 

 circuits. Mutual impedance or admittance between the two sides of 

 any circuit does not unbalance the circuit. Mutual impedance or 

 admittance, added between either of the given circuits and any third 

 circuit of the system, will not unbalance both of the given circuits. 

 This leaves admittance shunted directly from one given circuit to 

 the other given circuit and mutual impedance between the two circuits 

 as the only source of crosstalk. 



Let the admittances added between the two circuits be a, b, c, d 

 connected between conductors 1 and 3, 3 and 2, 2 and 4, 4 and 1, 

 respectively, where conductors 1/2 form one circuit and conductors 

 3/4 formi the other circuit. These admittances may be resolved into 

 the sum and difference of four admittances, as shown by the following 

 table : 



_ {a+b-\-c-\-d) , {a-{-b-c-d) (a-b-c-\-d) {a-h+c-d) 

 lto«5a- ^ -i- ^ -h ^ -h ^ 



3 to 26= + - - 

 2to4c= - - + 



4 to 1 ^ = - + - 



By the principle of superposition the effect of the given admittances 

 a, b, c, d will be practically the same as the sum of the effects of the 

 four component admittances taken individually. The first component 

 admittance (a + 6 -f c + rf)/4 is added symmetrically between the two 

 wires of one circuit and the two wires of the other circuit. This will 

 not disturb the symmetry of either circuit and will, consequently, not 



