PART 4. 



STRUCTURE AS DEFINED BY INTERLACINGS, HEADS, AND SEMIHEADS; 

 A COMPLETE CENSUS OF TRIAD SYSTEMS IN FIFTEEN ELEMENTS. 



By F. N. Cole. 



1. INTRODUCTION. 

 In forming triad systems in 15 letters, there are only four typical openings, viz: 



which, from the way in which the triads containing 1 are laced with those containing 2, may be 

 called the single tetrad, triple tetrad, hexad, and duodecad types, respectively. It turns out 

 in the present investigation that, witli a single exception, the tetrad tyije (single or triple) is 

 always present, so that in the final census only openings I and II need be considered. These 

 openings are then treated in sections 4, 5. 



2. THE DUODECAD OPENING. 



We show here that a triad system in the 15 letters can not be made up with duodecads 

 alone. To this end we note that opening IV above has the following group of 24 substitutions 

 which convert it into itseK : 



(1 2) (4 5 15 14 13 12 11 10 9 8 7 6),| 

 (4 6) (5 7) (8 15) (9 14) (10 13) (11 12)/ 



If the triads with 1 and 2 (excluding 12 3) are denoted by a, h, c, d, e,/and a', h', c', d', e',f', 

 respectively, this group is equivalent to 



{ iaf'fe'ed'dc'cb'ba'), (ai) {cf) (de) (b'f) (c'e') } 



and suffices to interchange the accented and unaccented letters with preservation of order of 

 sequence, to move each set of letters in a cycle, and to reverse the order of each set. 



If now the triads with 3 are laced through those with 1 and 2, in the duodecad manner in 

 each case, it may happen that these new triads (1) connect two successive ones of the 1 set or 

 2 set, or (2) do not exhibit such a sequence. It readily follows then, with the help of the group 

 above, that the only typical lacings are the following 14 : 



dbcdef abdcfe ahefcd 

 abfced 

 abfdec 



abcdfe 

 abcfed 

 abcefd 

 abcfde 



abdfec 

 abdfce 

 abecfd 

 abedfc 



acebfd 



the first 13 presenting the sequence ab, and the last one no such sequence. 



This last one, with no sequences, may be worked out in some detail as an example of the 

 method employed throughout this paper. We have to write down the triads with 3, lacing, 

 say, those with 1 in the order acebfd, and those with 2 without sequence. We can not use 3 4 8, 

 since this would give the sequence a' b' ; we must take 3 4 9, 3 5 8, or 3 5 9. These lead to the 

 four possibilities : 



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