•284 BELL SYSTEM TECHNICAL JOURNAL 



that for r a primitive root of a number N for which \{N) = <t>{N) and 

 in particular for a prime number iV 



rUN)i2 = _ 1 (modiV). 



This means that X(iV)/2 consecutive spHces with a spread r result in a 

 spread of minus one. It is readily shown that this in turn means that 

 there will be two conductor units No. b and No. 6 + 1 in the first length 

 of cable which ultimately will be extended to connect respectively to 

 units No. 6+1 and No. h. In Fig. 4 two units No. 6 and No. 7 meet 

 this requirement. To illustrate the use of this artifice it will be sup- 

 posed that a cable of 12 units is to be spliced. Referring to Fig. 4. for 

 guidance, the arrangement shown in Fig. 5 is set up readily. The 

 first two columns indicate the splicing assignment, and the succeeding 

 columns are then derived from these. The eleven units 1,2, •••,5,6, 

 8, 9, • • • , 11, 12 are assigned exactly in conformity with the scheme of 

 Fig. 4, ignoring the break in sequence between No. 6 and No. 8. 

 Unit No. 7 is then simply spliced to itself throughout. 



Fig. 5 



Undoubtedly there are other equally acceptable artifices for extend- 

 ing further the practical scope of the simple results. The prime num- 

 bers and the prime numbers plus one constitute nearly fifty percent of 

 all numbers in the range in which the practical worker is likely to be 

 interested, however, and when it is borne in mind that normally he has 

 latitude in his choice of N it is seen that the material here presented is 

 adequate for his needs. 



The writer is indebted to Dr. D. H. Lehmer for pertinent suggestions. 

 The entire treatment for the case of numbers for which X(iV) < <t>{N) 

 follows a line of attack suggested by Mr. Marshall Hall, and but for his 

 helpful interest this presentation would have been lacking in formal 

 completeness. 



