22 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv. 



The first 7 elements, as we have seen, must constitute by themselves a triad system A?, 

 while each one occurs in 4 additional triads with pairs of letters from the last S. Of these 8 

 letters aU possible pairs occur, 4 self-conjugate and the rest in 12 conjugate pau-s of pairs; 

 28 in aU, so that every such pair is joined in a triad with one of the digits. Two pau-s that are 

 conjugate must form triads with the same digit as third element. Hence the 4 self-conjugate 

 pairs, like Aa, are either all completed to triads b}' the same numeral, as 1, or else by two 

 numerals, as 1 and 2, each joined with two pairs. Triads not self-conjugate, as SAB, Sat, occur 

 two by two. 



Assembling in colunms of four the pairs associated with the several digits, we shall have 

 a seven-by-four array. We shall find that there are five types of such arrays, aside from per- 

 mutations of entire columns. To complete them to triad systems, it remains only to annex 

 a triad system A? constituted upon the seven digits. This can be done in, a variety of ways, 

 so that several systems will result from each seven-by-four array. Triads composed of one 

 digit and two letters shall be termed mixed. First we tabulate the mixed triads, writing down 

 the pairs of letters only. 



Pairs from Mixed Triads, Class VII 1. 



Here explanation is necessary. Any column could be selected as the second, whence the 

 thii'd would follow. Beside the exchange of conjugate letters in independent pairs, there are 

 still permissible the substitutions 



{AB){ab), {OD){cd), {AC)(,BD){ac){bd), (AD) (BC) (ad) (be), 



this last a result of the others. Compared with the second or the third, any later column may 

 be either cross-tied or not. For example, the fourth is cross-tied to the second by the triads 

 2AB, 2 CD in the one and 4 AC, ABB in the other; hence also by the remaining pairs in the 

 two columns. Notice also that when it is cross-tied to the second column it is necessarily cross- 

 tied to its cognate column, the third. As an example of the opposite kind, the columns 6 and 7 

 in class VII2 are not cross-tied to columns 2, 3, 4, or 5. 



If all four self-conjugate pairs stand in a single column, there are but two nonequivalent 



classes of seven-by-four arrays, those having the other six columns all cross-tied, 



and those having four aU cross-tied and the two others not cross-tied with them. 



All others having column 1 can be reduced to either VIIl or VII2. 



In the other alternative, when self-conjugate pairs of elements appear in two colunms, two 



in each, there are three classes of schedules. Let the columns containing self-conjugate pairs 



be the first and second ; the other two pau-s in the first column are cross-tied to the self -conjugates 



in the second, and -vice versa. Therefore also there wiU be another column — ^let it be taken for 



the third — cross-tied to both the fu'st anil the second. Compare the four subsequent columns 



with the third. Either 4, 2, or are cross-tied with this third. If two are not, select them for 



the sLxth and seventh columns. The resulting arrays are the following: 



