566 BELL SYSTEM TECHNICAL JOURNAL 



2y{a - 6 + c- • • ) - 27'(a' - 6' H ) + ^' (a' - i' + • • • ) 



X 



2ys — 27V + --I y^s^ 



But the sum of alternate intervals in the transposition periodic 

 section 5 must equal 5/2. Therefore a — b -\- c- • • — a -\- {c — h) 

 -\- {e — d) -\- • • ' = 5/2, and to the same approximation, 



(fa + ^O' 1 - [1 - 2t-x + 2tV] 



£ / Y 

 27 



1 - ^ (a2 - fe2 + . . .) + tl (^3 _ ^3 ^ . . .) 

 ^ 1 - 75 + 27V/3 - • • • J 



£/7_ Z^\ Ff 2x 2{a? -h''+ '-') \ 



1 2x , 2^2 ^ / , 2x \ a2 _ ^,2 ^_ . . . 



+ 



13 5+52 \^ s ) 



+ 4(«^-^^+ --oIt^] 



353 



It will be noticed that the crosstalk varies linearly with the distance 

 from the first transposition, approximately, and that by a suitable 

 choice for this distance the crosstalk may be reduced to zero to the 

 first approximation. This is, however, not a matter of especial 

 practical importance, for incidental irregularities contribute to the 

 crosstalk in practice. As soon as the crosstalk due to the regular 

 transposition system is reduced to the order of that due to the acci- 

 dental irregularities further reduction of this crosstalk is not a matter 

 of commercial importance. The accidental irregularities in the dis- 

 tribution of the wires therefore set a limit to the extent to which it is 

 worth while to reduce the length of the transposition sections. 



If there are but two transpositions in the periodic interval a = s/2, 

 b = c ^ ■•■ = 



2 116^4/^2 



1 2jc \ X / . 2x\ 1 . . 



— r)~^(l — T I y^ \ approximately. 



2 s / s \ s 



If X = 5/4, I vanishes as to first order terms. If x = s/2 or 0, / has its 

 maximum value, , 



E/ Y .Z_\ s 

 16 "^ 4F 



