An Application of Number Theory to the Splicing of 

 Telephone Cables * 



By H. P. LAWTHER, JR. 



The consideration of a simple and practical splicing scheme for mini- 

 mizing the recurrence of same-layer adjacencies among telephone circuits 

 in long cables leads to a problem in Number Theory whose solution calls 

 for some extension of the previous work in this field. The solutions for 

 numbers not greater than 139 have been computed, and a table of these 

 is included. 



SOME time ago in connection with the placing of a long telephone 

 cable the writer had occasion to attempt 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, super- 

 ficially 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. That speculation upon an art so 

 mundane as that of telephone cable splicing should have led to a propo- 

 sition in the oldest and 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 con- 

 tained within a cylindrical sheath. The conductor units are either 

 pairs of quads of wires. The layers are one unit in thickness, and suc- 

 cessive 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 that in an unbroken length of cable any one con- 

 ductor unit will experience shoulder-to-shoulder adjacency throughout 

 this distance with the two conductor units lying on either side in the 

 same layer, and its experience with these two conductor units will be 

 unique. Cables usually are manufactured in uniform lengths of from 

 750 to 1000 feet, and a longer cable is made up from a succession of such 



* Published in Amer. Math. Monthly, February, 1935. 



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