No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 23 



Pairs from Mixed Triads. 



Class VIIS is reducible to class VII2. The array of VII2 has the column 1 unique, cross-tied 

 to (interlaced with) all the others, a character not found in anj' other column. In the array of 

 class VIIS, the column S is unique in the same particular. From this clue, one finds without 

 difficulty a transformation of the latter array into the former. This transformation does not 

 preserve, however, the pairs of conjugate letters; in other words, it does alter the substitution 

 S with reference to which the systems are constructed ; but it changes S into another substitu- 

 tion S' of the same type (1)'(2)^ The transformer is this: (A) {B d b a D c C) (16 5 4 2 7 3). 

 Since the array of a system of class VIIS with respect to a substitution S, of type 

 (1)'(2)^, can be transformed into an array of class VII2 with respect to a different 

 substitution S' of the same type, all systems belonging in the one class belong also 

 in the other; and hence class VIIS does not require a separate investigation. 

 Upon these arrays we are now to superpose triad systems, Ay's, constructed in all non- 

 equivalent modes from the seven digits. First, for the class VII 1, there is an immediate 

 deduction available. The columns are triply cross-tied (interlaced), so that they indicate an 

 inherent triad-system or Aj. Compared with this inherent system, the A, to be imposed must 

 have 7, S, 1, or triads in common. As no column and no inherent triad is unique in this 

 array, no further distinction is possible, and there are precisely four essentially different systems 

 in this class. 



Supplementary Sets, Ay's, fob Class VII 1. 



System VIII,: 123, 145, 167, 246, 257, 347, 356. 

 Sj'stem VIII,: 123, 145, 167; 247, 256, 346, 357. 

 System VIIl^: 123, 146, 157, 247, 256, 345, 367. 

 System VIIls: 124, 136, 157, 237,256,345,467. 



In the array for class VII2, as has been pointed out, column 1 is unique, and the two columns 

 6, 7 are unlike 2, 3 and 4, 5 in relation to cross-tying or interlacing. AU three of these pau's, or 

 else only one of them, or none at all, may be united with numeral 1 in the superimposed Aj. 

 If only one, that one may be either 2 3 or 6 7. There are thus four cases, and each can be com- 

 pleted in two ways, giving apparently eight supplementary sets of triads or Ay's. 



