No. 2) TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 15 



must occur, we see that ad and be must occur in triads with a double letter, so that either ac 

 or hd, not both, will also occur in such a triad. There are accordinj^ly two possible schedules: 



Case 4: Triads ABC, ddd; Aab, Acd, Bab, Bed; Cac, Cbd; aad, bbc, cca; abd, bed. 



It is noticeable that in this arrangement no two of the letters abed are interchangeable; 

 the same is true in the next, the final case. 



Case 5: Triads ABC, ddd; Aab, Acd, Bab, Bed; Cac, Cbd; aad, bbd, cob; abc, acd. 



After the above distinction of five schedules there is for each case a subdivision into species 

 by means of the pairs of subscripts attached to letters in the triads with A, B, and C. The 

 numbers of such species, a priori, are for case 1, fom*; for case 2, four; for case 3, three; and 

 for cases 4, 5, four each. Some of these are realized by two completed systems, some by one, 

 or by none; so that, in all, 26 systems apparently distinct are found invariant under an operation 

 or substitution (1)^(3)^ 



Species in case 1. — In case 1 the sets of letters a, b, c, d are indistinguishable, and are dis- 

 tributed to the exchangeable letters A, B, C in all the complementary pairs. As a means of 

 abbreviating tabulation, write these triads in the order 



Aab Acd 

 Bac Bbd 

 Cad Cbc, 



and instead of rewriting these letters with subscripts, write the subscripts only. On each letter 

 independently the subscripts may be changed if we do not change the cychc order 123; and this 

 order may be reversed to 132 for aU four sets of letters simultaneously. The first three pairs of 

 subscripts may be fixed arbitrarily as 1, 1, thus: ajji, c^d^, a^Ci. We need only write the three 

 remaining pairs in their relative positions, without letters: 



(a) 

 11 



11, m 

 11 11 



11, (7) 

 12 12 



11, (S) 

 12 11 



12 

 13 



These four cases exhaust the possibilities; for we can reduce aU others to these four by permis- 

 sible interchanges of letters and subscripts. There should be nine cases where indices 2, 3 do 

 not appear in the first row or the second and we see that (a) represents one, (/3) four, and (7) 

 four. If the pair 1 2 or 1 3 occurs in the second line, the reduction is less obvious, but not 

 intricate, as one example will show. Let these six pairs in same order be afii, c^d^; aiC^, b^d^; 

 a^dj, biCj (representmg also, of course, their 12 conjugate pairs). Write subscripts only, and 

 change those of d cychcally by writing 1 for 2, etc. This leaves 



11, 13; 11, 11; 11, 13. 



Next, c is written for d and vice versa, as the letters are of equal significance; then for 31 write 

 12, one of its conjugates. Now we have 



11, 12; 11, 13; 11, 11, 



since second and third lines have exchanged subscripts. But this is case (5) by permutation of 

 letters A, B, C. By such verification we confirm the completeness of tliis list of four species. 



Species in case 2. — In case 2 tliere is no distinction between letters b and c, but a and d are 

 not interchangeable with them or with each other. First we fix triads Ad^a^, Bdfii, Cd^c^, so 

 that all indices are of determinate meaning except for a choice between the orders 123 and 132. 

 Since the combination bed is to occur in two sets of three, these can onl}' be biC/l^ and biCjd^; 

 accordingly the triads Abc must include AbiC^. The pairs with B and C admit some freedom 

 of choice still, all alternatives being reducible to these four following: 



(a) CaJ), m CaJ), (7) Ca,6j (5) Cafi^ 

 BaiCi BttiC^ BuiCj Ba^c^ 



Species in case 8. — Letters a, b, c are not yet distinguishable separately in case 3. Fix 

 first, as in case 2, the triads Adfii, Bdfi^, Cd^Ci. Next, two sets of triads are to contain abc, 



