24 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv 



Supplementary Sets, Ay's, for Class VII2. 



System VII2, 1 ir.7-p46, 257, 347, 356. 



(System VII22)J ' ' '1247, 256, 346, 357. 



System ¥112, !,.,„ . „ ,4fi.!245, 267, 347, 365. 



(System VIl2,)j ' ' '1247, 265, 345, 367. 



System VII2,1 (236,456,247,357. 



(System VII2,)J ' ' '1237, 457, 246, 356. 



System \qi2, 1 ifi,.f237, 453, 674; 256. 



(System VII28)J ' ' '1235, 456, 672; 347. 



These are equivalent, two and two. For systems VII2, and VII22 the transformer is 

 obviously {67){AB){ah){CD){cd). The same transformer relates VII25 and VIU^. SUghtly 

 more intricate is the transformation of VII27 and VIl2g, viz by the substitution (24) (35) (67) 

 {ABDOiahdc); and that which exchanges VII23 and VII2„ namely (23) ii5){Q7 )iABab){DCdc). 

 We omit, therefore, the even-numbered systems in the above list, as indicated by parentheses. 



In the array for class VII4, column 3 is imique, and the others are paired by being cross- 

 tied or interlaced. The three pairs, 12, 45, 67, are distinct or imlike; for the first are triply 

 interlaced with cohunn 3 by sets of foiu- letters, the second are interlaced with each other, and 

 each by itseK vnth column 3, while the last pair are interlaced with each other but not with 

 colunm 3. It is important to observe that we can exchange simultaneously the members of all 

 three pau's, by the substitution (ADad) (BCbc) (12) (45) (67). Tliis allows us to omit one of 

 every two that have one of these pairs of numerals in a triad with 3, as for example 345. 



Supplementary Sets, Ay's, for Class VII4. 



System VII 4,: 312, 345, 367; 146, 157, 256, 247. 

 System VII 4^: 312, 346, 357; 145, 167, 247, 256. 

 System VII 43: 316, 345, 327; 142, 157, 652, 647. 

 System VII 4,: 314, 325, 367; 126, 157, 427, 456. 

 System VII 4^: 314, 326, 357; 125, 167, 427, 465. 

 System VII 4,: 314, 326, 357; 127, 165, 425, 467. 



This list is complete. For we need only consider the triads containing the element 3. 

 Either aU tliree contain pairs whose columns are interlaced in the array, (VIl4i), or only one, or 

 none. Tliat should give us (l+6-l-4 = )ll systems, after making allowance for the automor- 

 pliism mentioned just before the list. A fm-ther reduction is effected by observing that each of 

 the thi-ee operations like {AB){CD){ab){cd) exchanges each of two pairs of columns, as (45) (67), 



leaving the other colunms of the array unaltered. Accordingly the five systems VII42 



VIl4o represent ten, and VIl4i makes up eleven, the fuU coimt. 



•The array for class VII5 has the unique column 3, the imique pair of cohmins 1, 2 containing 

 conjugate pairs, and the interchangeable pairs of columns 45, 67. We shall take account of five 

 substitutions among letters in the array, and their effect in permuting columns and their 

 respective digits. 



r,: (^B) (06) produces (47) (56). 



T,: {CD){cd) produces (46) (57). 



7;: iAB){CD){ab){cd) produces (45) (67). 



T,: {AC){BD){ac)ibd) produces (12)(67). 



T,: (AD) (BC) (ad) (be) produces (12)(45). 



Hence we distinguish only four cases, different as regards the pairs associated in triads with 

 the unique numeral 3. Either all the pairs 12, 45, 67, or the pair 12 only, or one of the others 

 exclusively, as 45, or none of them, must occur with 3. Each of these admits evidently two 

 modes of completion, but two of the resulting eight systems are redundant, as wiU be explained. 



