274 BELL SYSTEM TECHNICAL JOURNAL 



lengths spliced end-to-end. At each splice point a large number of 

 different splices is possible among conductor units. In general, wire- 

 to-wire splices are not made, and considerable mixing up is achieved. 

 For reasons which need not be given here it is considered desirable 

 from the standpoint of crosstalk control that each telephone circuit 

 experience the minimum amount of same-layer adjacency with every 

 other telephone circuit. 



For the purposes of this discussion it will suffice at present to con- 

 sider the cross-section of a cable as a simple closed sequence of N con- 

 secutively adjacent units. As an example, the array presented by a 

 circular picket fence would be of this character. Each conductor unit 

 in a cable is identifiable, and it will be assumed that each has been 

 "tagged" with one of the numbers 1, 2, 3, 4, • • • , A'^ in such sequence 

 that units bearing consecutive numbers lie adjacent — remembering 

 that unit No. 1 and unit No. N also lie adjacent. While this simple 

 picture of the cable cross-section is representative truly of only a single 

 layer structure, still the results of a study of it may be fitted to apply 

 to practical cases. Schemes for accomplishing this will suggest them- 

 selved to the practical worker, and their discussion here would burden 

 this presentation unduly. 



Consider now two consecutive lengths in a completed cable and focus 

 attention upon a conductor unit in one of these. At the splice point 

 this conductor unit may connect to any one of the conductor units in 

 the second length, and the two conductor units which lie alongside the 

 latter in the same layer in the second length may connect to any two of 

 the A^ — 1 remaining conductor units in the first length. As an ex- 

 tended conductor unit traverses the completed cable, then, it may ex- 

 perience same-layer adjacency successively with any possible combina- 

 tions two at a time of the other extended conductor units, and in any 

 order, sequence, or repetition of these as determined by the splicing 

 scheme that is used. Since there can be but \_{N — l)/2]* totally 

 different combinations two at a time of A — 1 different objects it is 

 evident that \_{N — l)/2] successive cable lengths is the maximum 

 possible number for an extended conductor unit to traverse without 

 incurring repetition of at least one of the same-layer adjacencies that 

 occurred in the first of these lengths. 



Any splicing scheme that is devised for practical use must embody 

 the utmost in simplicity. For this reason it is considered highly 

 desirable (1) that the required results be achieved through repetition 

 of the same splicing instruction at consecutive splice points, and (2) 



* The symbol [xjy] means the greatest integer not greater than xjy. 



