An Application of Number Theory to the Splicing of Telephone Cables

O M E time ago in connection with the placing of a long telephone cable the writer had occasion to a t t e m p t the specification of a splicing scheme designed to minimize the recurrence of same-layer adjacencies among the telephone circuits as they threaded their way through successive lengths of the completed cable. The task, superficially so simple, proved to be one of most intriguing difficulty, and the pursuit of the solution led a confused investigator stumbling into the province of number theory. T h a t speculation upon an art so m u n d a n e as that of telephone cable splicing should have led to a proposition in the oldest a n d most neglected branch of mathematics seemed to be especially worthy of note, for few applications so practical have been found. In the course of the investigation certain small ground apparently was covered for the first time. It was felt, therefore, that the story would be of passing interest alike to the mathematician and to the engineer. The present standard cables for long distance telephone service are manufactured as a series of concentric layers of conductor units contained within a cylindrical sheath. The conductor units are either pairs of quads of wires. The layers are one unit in thickness, and successive layers either spiral in opposite directions of rotation, or in the same direction but with different pitches. The feature of importance to this discussion is t h a t in an unbroken length of cable any one conductor unit will experience shoulder-to-shoulder adjacency throughout this distance with the two conductor units lying on either side in the same layer, a n d its experience with these two conductor units will be unique.