76 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vot. xiv 



With a semihcad 3 4 7, 3 5 8 there are 17 typical sets of 3-triads: 



3 4 7 3 5 8 3 6 9 3 10 12 3 11 15 3 13 14 



3 10 13 3 11 15 3 12 14 



3 10 14 3 11 13 3 12 15 



3 10 15 3 11 13 3 12 14 



3 6 11 3 9 10 3 12 15 3 13 14 



3 9 13 3 10 15 3 12 14 



3 9 14 3 10 13 3 12 15 



3 9 15 3 10 13 3 12 14 



3 6 13 3 9 10 3 11 15 3 12 14 



3 9 11 3 10 15 3 12 14 



3 9 14 3 10 12 3 11 15 



3 10 15 3 11 12 



3 6 15 3 9 10 3 11 12 3 13 14 



3 11 13 3 12 14 



3 9 11 3 10 13 3 12 14 



3 9 13 3 10 14 3 11 12 



3 9 14 3 10 13 3 11 12 



These lead to only three triad systems with semihead but no head or triple tetrads. 



There now remains only the case of no triple tetrads, heads, or semiheads. The first 

 3-triad may be taken as 3 4 8, and the opening set of 14 triads is unchanged by the substitution 

 s=(l 2) (5 6) (9 10) (11 12) (13 14) only. The triad 4 15 a; has x = 7, 9, 10, 11, or 12, arid under 

 s 9 is equivalent to 10 and 11 to 12. We have then three typical cases: 4 7 15, 4 9 15, 4 11 15. 

 For the first of these 8 15 a; gives x = 5 or 11, and the continuation is not unreasonably long. 

 But the cases 4 9 15 and 4 11 15 each subdivide into foiu" cases instead of two, making the total 

 labor five times that of the 4 7 15 case imless some method of compression could be devised. 

 Fortimatcly such a method was at hand. 



The triad pairs 4 1 5, 4 2 6 and 7 1 6, 7 2 5 exhibit a tetrad. If, then, a triad system has 

 been constructed so far as to show its triads with 1, 2, 4, and 7, it may be possible by inter- 

 changmg the pairs 1, 2 and 4, 7 to throw this system into one already identified. For example, 

 arriving at the set 



4 3 8 4 9 15 4 7 11 4 10 12 4 13 14 

 7 3 9 7 8 13 7 10 14 7 12 15 



we find that the substitution (1 4) (2 7) (3 15 12 9 13 11) (8 14 10) throws this into a set with 

 the original 1-triads and 2-triads, but containing 3 4 8, 4 7, 15 and therefore coming under a 

 case already explored. 



As another example, the set 



4 3 8 4 11 15 4 7 10 4 9 13 4 12 14 

 7 3 11 7 8 13 7 9 14 7 12 15 



is thrown by (1 7 2 4) (5 6) (3 9 11 12 15 13 10) into the case 4 3 8, 4 9, 15. In fact, the nearly 

 150 cases under 4 11 15 reduce to only 6 by this process. 



In the end there are found to be 15 triad systems with no triple tetrads, head, or semihead. 



6. SUMMARY. 



The following table gives the number of triad systems of each type for 15 letters: 



Triple tetrad and head 22 



Triple tetrad, no head, but semihead 12 



Triple tetrad, no head, or semihead 26 



No triple tetrad, but head 1 



No triple tetrad or head, but semihead 3 



No triple tetrad, head, or semihead 15 



No tetrad (Heffter's system) 1 



Total number of types 80 



