No. 2.] 



TRIAD SYSTEMS— WHITE, COLE, CCIMMINGS. 



85 



In Part 4 of this paper Mr. Colo disliiiKuishos four varieties of interlacing of duads in a 

 system, namely, tlio single tetrad or oktad, the tri])]o tcitrad, the liexad, and the dodekad. 



Thi«e ty])es correspond to what wo have designated in Part 3 as two-colimin or contracted 

 indices. Tlie triple tetrad, the oktad, the hcxad, and the dodekad corresponding, respectively, 

 to the indices 2'; 2, 4; 3=; 6. 



Since these four types of interlacing form the basis for Mr. Cole's derivation of tlie 80 

 systems, it seemed probable that the two-column indices might fui-nish a sufficient and unique 

 characterization for a triad system. 



The two-column indices for each of the 80 systems were, therefore, determined and are 

 exhibited in the following Table 2 : 



Table 2. 



System. 



ni.\.. 



IIIB.. 

 lUC. 

 HID.. 

 UE... 

 IV A.. 

 IVB.. 

 VD... 

 VA... 

 VC... 

 VB... 

 II A... 

 VID.. 

 IID... 

 IIC... 

 IIB... 

 HF... 



15 



9 



V,47l 

 VIB.. 

 IC... 

 V,4al 

 VIC. 



11 



10 



VI, 335 



System. 



4 



7 



IB 



VIA.... 



12 



VI,2,J.. 



13 



VI, 2,7. 



6 



3 



1 



VI, la^.. 



26 



28 



V,l>... 



25 



V,4t2.. 



14 



8 



5 



27 



35 rc<)lel 



32 



VI,2,a. 

 21 



vi,'2i«;.' 



2 



System. 



31 



24 



20 



19 



36 [Cole]. 

 lA 



V,4/31... 



V::::::. 



22 



VI,3,Y.. 



II, l7 



34 [Cole] . 

 V,4(32-.. 

 V,4J1.... 

 VI, 3w.. 



23 



VI,3jT.. 



33 



II, Ij 



18 



29 



16 



30 



II, I3 



VII 



68 

 72 

 74 

 77 

 81 

 S6 

 65 

 70 

 66 

 69 

 74 

 69 

 70 

 77 

 77 

 66 

 73 

 74 

 76 

 78 

 74 

 74 

 76 

 79 

 84 

 90 



This table shows that the two-column inchces suffice to estabUsh the noucongruency of 

 72 of the 80 systems, but fail to distinguish uniquely the remaining 8 systems. The systems 

 not uniquely determined by their two-colimm indices consist of the five pairs of headed sys- 

 tems IIC, IID; IIB, IIF; HE, IVA; IVB, VD; VA, VC; one pair wdtli a group but no head 

 V451, V4;82; and two pairs of grouplcss systems 18, 29; 32, 35 [Cole]. 



Perfect discrimination is possible by the use of a double entry table, 15 by 15, shoAving 

 not merely the number, but the exact distribution of triple tetrads, oktads, hexads, and 

 dodekads in each of the eight pairs of apparently duplicate sj'stems. 



llie investigation for one of these pairs of apparently duplicate systems, IIF and IIB, 

 is given below in some detail. 



The system IIF is arranged in a 15-by-7 alTa3^ Each element heads one column; below it 

 are placed the seven duads of elements that occur with it in triads of the system. The two- 

 column indices for the 105 pairs of columns are determined, and we find that the indices 2'; 

 2, 4; 3-; 6 belong, respectively, to 11, 42, 0, and 52 pairs of columns. 



To exhibit more evidently the types of intcrlacings existing amongst the duads in the 105 

 pairs of colunms, we now arrange the two-column indices for the system IIF in the following 

 15-by-15 array of Table 3. 



