276 BELL SYSTEM TECHNICAL JOURNAL 



will be described as made with a "spread of one," a "spread of two," 

 of a "spread of three," respectively. It is readily shown that for a 

 spread number 5 to be applicable to cable of N units it is necessary and 

 sufficient that s be prime relative to N. 



Figure 4 shows the splicing of six pieces of eleven-unit cable through 



1 -^ 1 -^ 1 ^ 1 -> 1 -> 1 



2^ 3-^ 5^ 9 ^6->ll 



3-^ 5-^ 9^ 6-^ 11 ^10 



4-^ 7-^ 2-> 3-^ 5^ 9 



5^ 9^ 6^ 11 ^10-^ 8 



6-^ 11 -^10^ 8-^ 4-> 7 



7-> 2-^ 3^ 5^ 9-^ 6 



8^ 4-^ 7-^ 2-^ 3^ 5 



9-^ 6-^ 11 -^10^ 8^ 4 

 10^ 8^ 4^ 7-> 2^ 3 

 11 ^10^ 8^ 4-> 7-> 2 



Fig. 4 



the successive application of five consecutive identical splices, each 

 with a spread of two. Following the "key" of the first and second 

 columns, the succeeding columns are written down immediately. 

 Scrutiny of the sequences of numbers appearing in the several columns 

 reveals at once the fundamental properties of the spread. For a 

 cable of N units these are : 



1. Successive applications of a spread of 5 for n times result in a 

 spread of s". 



2. A spread of minus 5 is equivalent effectively to a spread of plus s. 



3. A spread of KN -{- s {K is an integer: positive, negative, or zero) 

 is the same effectively as a spread of s. 



The problem of achieving the minimum possible recurrence of same- 

 layer adjacencies among conductor units through the application of 

 successive similar splices in accordance with a simple spread now may 

 be stated formally in the terminology and symbols of number theory. 

 If N, an integer, is the number of conductor units in the cable, and if s, 

 an integer prime to N, is the spread number used, then it is required to 

 find a value for 5 for which the companion relations 



5^^ = ± 1 (mod TV), 



s^ ji ±\ (mod TV), b <d 



determine the largest possible integer d. 



From the foregoing introductory discussion it should be noted that 



