86 



MEMOIRS NATIONAL ACADEMY OF SCIENCES. 



Table 3. 



[Vol. XIV. 



In this table the two-column indices 2'; 2, 4; 3^; 6 are replaced, for the sake of brevity, by 

 the single figures 2, 4, 3, 6, respectively. The figure placed at the intersection of any row with 

 any column shows the index for the pair of colimins formed from the two elements which lead 

 the row and head the column, respectively. For example, in Table 3, the indices for the pairs of 

 columns he, Id, hi are 2^; 2, 4; 6, respectively. 



In the 15 by 7 rectangular array for the system IIF, the column headed by the element a 

 may be united with each of the columns headed by one of the remaining 14 elements to form 14 

 pairs of colunuis. The figures tabulated under the element a in table 3 shows that of these 14 

 pairs of columns there are 4 pairs with the index 2^; 2 pairs with the index 2, 4; and 8 pairs 

 with the index 6. Interpreted in terms of interlacings these two-column indices place in 

 evidence the fact that the duads of column a arc united with the duads in the remaining 14 

 columns by 4 triple tetrads, 2 oktads, and 8 dodekads. Similar results for each of the 15 

 elements are briefly summed up in the following Table 4, which we shaU designate as the table 

 of interlacings for the system IIF. 



Table 4. — Interlacings for system TIF. 



Since only those elements whose duads are similarly interlaced through a system may 

 belong to the same set of transitive elements, Table 4 shows clearly that the possible sets of 

 transitive elements for the system IIF are a, he, d, eg, f, 1367, 2458. A fact in exact accord- 

 ance with the resiilts obtained previously in the examination of this system by the method of 

 trains and also by the method of the thi'oe-column indices. 



The interlacings for the system IIB are exhibited in Table 5. 



Table 5. — Interlacings for the system IIB. 



A comparison of Table 4 mth Table 5 demonstrates conclusively the noncongruency of the 

 systems IIF and IIB. Table 4 shows that there is a column d in the system IIF the interlacings 

 of whose duads with the duads of other columns in IIF are represented by the numbers 2, 8, 

 and 4, corresponding, respectively, to two triple tetrads, eight oktads, and four dodekads. Table 

 5 shows no column in IIB with similar interlacings, therefore these two systems IIB and IIF 

 are certainly incongruent. The reader will observe other distinctive columns in these two 

 tables. 



