196 BELL SYSTEM TECHNICAL JOURNAL 



segments are called the plus segments, the far-end crosstalk is pro- 

 portional to the difference of the sum of the plus segments and the 

 sum of the minus segments. This difference may be called the 

 unbalanced length and the output-to-output far-end crosstalk is this 

 length multiplied by the far-end coefficient and by the frequency. 



If the sum of the plus segments equals the sum of the minus seg- 

 ments, the unbalanced length will be zero. The poles of a line are 

 necessarily spaced somewhat irregularly but for a single circuit 

 combination the unbalanced length could be made very small by 

 carefully picking the transposition poles so as to keep the sums of the 

 plus and minus segments about equal. This procedure is impractical, 

 however, because many circuit combinations must be considered and 

 because necessary line changes would prevent the maintenance of very 

 low initial unbalanced lengths. 



In practice, therefore, the segment lengths are allowed to deviate 

 in a chance fashion from the mean segment length. The unbalanced 

 length varies among the various circuit combinations depending on the 

 arrangement of the transpositions which determines the order in 

 which plus and minus segments occur. For any particular combina- 

 tion, the unbalanced length has a wide range of possible values and 

 its sign is equally likely to be plus or minus. 



In any transposition section, the length of any segment may deviate 

 from the average segment length for that section. If the sum of the 

 squares of all the deviations in each transposition section is known, 

 the unbalanced length for a succession of transposition sections may 

 be estimated, that is, the chance of the total unbalanced length lying 

 in any range of values may be estimated. 



Letting ^'i^ be the sum of the squares of the deviations for the first 

 transposition section, etc., and letting R be the r.m.s. of all the possible 

 values of the total unbalanced length in all the sections, the following 

 approximate relation may be written: 



i?2 = 5^2 _^ 5,2 _^ . . . etc. 



The chance of exceeding the value R may then be computed. For 

 example, there is about a one per cent chance that the total unbalanced 

 length will exceed 2.6R. 



In making rules for locating transposition poles the first step is to 

 determine a value for R. For example, if consideration of tolerable 

 crosstalk coupling indicated that there should not be more than one 

 per cent chance that the total unbalanced length in a 100-mile line 

 would exceed one mile, then R, the r.m.s. of all possible values of the 

 total unbalanced length, should not exceed 1/2.6 miles. Since R is 



