NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 25 



Supplementary Sets, At's, for the Class VII5. 



System VII5. | ,^7.1146, 157, 247, 256. 



System VIIS^ f ' ^ ' 'll47, 156, 246, 257. 



System VII53 Qr:7.1145, 167, 247, 265. 



(System VIl5,)r^^' '^^^' "^^'il47, 165, 245, 267. 



System VII55 

 System VII5, 



316,3^5,327; 



(142, 157, 647, 625. 

 147, 125, 642, 657. 



System VII5,. 1 (125,167,427,465. 



(System VII5,). J ' "^ ' "^ '1127, 165, 425, 467. 



In this list two equivalences can be detected. A distinction has been pointed out, in the 

 array of class VII5, between the set of columns, 123, and the others, columns 45 67. Substitu- 

 tions can be seen which will transform any one of these latter columns into itseK, permute 

 the tlu-ee others in cycle, and permxite cyclically also the first three columns. Select the pair 

 ACiroTQ column 4. Under 5, 6, 7 note the pairs containing A and C; similarly under 132. 



The substitution (A)(0) (abB) (cDd) is found to convert column 4 into itself, hence it is equivalent 

 to the operation on numerals : 



(4) (132) (567), ■ 



and this transforms the supplementary system ¥115^ into VII 5,. In the same way is found the 

 operation : 



(B) (D) (dcO) (bAa)=(5) (123) (467), 



which shows an equivalence between VII55 and VII59. 



Four other relations, found by the sequence method of Miss Cummings, are readily verified. 



System VIl2,=System VIIU, by the substitution (<dl)(Dd). 



System VIl4,^System ¥1123, by the substitution (23)(4c)(5C)(6d)(7Z?). 



System VIl5i^ESystem VII25, by the substitution 



1234567 AaBb Oc Dd^ 



f 1234567 Aa Bb Oc Dd\ 

 \l7QbOBc 25 43 aD Ad) 



System VIl53^System VIII 4, by the substitution 



1234567 Aa Bb Oc Dd^^ 



/1 234567 Aa Bb Oc Dd\ 

 \213ABba OD cd 46 75/ 



Accordingly there remain as distinct systems in this section the following 16: 



Class VIIl, systems „ 2, 4, oi 



Class VII2, systems 3, 5, ,; 



Class VII4, systems 1, 



Class VII5, systems 



No proof of nonequivalence has been given, save in a few special cases. It is beheved that 

 aU redundances have been ehminated from each separate section. To Miss Cummings wdl fall 

 the discovery of equivalences in different sections, and the proof of essential difference in the 

 residue. As a summary result, we know that the 71 triad systems analyzed and listed in the 

 foregoing include certainly all that admit any substitution other than the identity. 



II 3! 4) 5> e> 

 2> 5) 61 7- 



