NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 75 



Wc find that the x of 4 3 a; must be 7, 8, ... , 15, and that 8, . . . , 15 are equivalent. The 

 typical cases are then 4 3 7 and 4 3 8. Of these 4 3 7 has a group of order 128, composed ol 

 those substitutions of the group of order 768 which leave 4 and 7 unchanged or interchange 

 them. With the aid of this group of order 128 we find that we may take as typical cases 

 3 5 6 or 3 5 8. 



The case 3 4 7, 3 5 6 exhibits a triad system in the seven letters 1,2, 3,4,5, 6, 7, included 

 in the triad system in the 15 letters. We then speak of the latter as having a 7-head, or simply a 

 head. The triads with 3 are now found to be the six followmg sets: 

 347 356 38 11 



3 8 12 



Working out their continuations, wc find the 22 triad systems with triple tetrads and 7-head 

 exliibitcd ui the previous chapters of this memoir. 



The case 3 4 7, 3 5 8 presents what one may perhaps be permitted to call a semihead. 

 The triads with 3 are here: 



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



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



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



3 9 13 3 10 14 3 11 15 



3 10 15 3 11 14 



3 9 14 3 10 13 3 11 15 



3 10 15 3 11 13 



Continuing these, we find 12 triad systems with triple tetrads and semihead. 



In the absence of cither head or semihead, the 3-triads form one of the following sets: 

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



3 10 15 3 11 14 



3 7 14 3 10 13 3 11 15 



3 10 15 3 11 13 



3 7 15 3 10 13 3 11 14 



3 10 14 3 11 13 



3 5 10 3 7 15 3 9 13 3 11 14 



3 5 11 3 7 15 3 9 13 3 10 14 



3 9 14 3 10 13 



3 5 15 3 7 11 3 9 13 3 10 14 



and these lead to 26 triad systems with triple tetrads but no head nor semihead. 



5. ONLY SINGLE TETRABS. 



The group of the single tetrad opening is of order 64 : 

 {(4 5) (6 7), (4 6) (5 7), (4 7) (5 6); 

 (8 11 15 12) (9 10 14 13), (8 9) (10 12) (11 13) (14 15); 

 (1 2) (5 6) (9 10) (11 12) (13 14)}. 

 This group also leaves the triad pair 3 4 7, 3 5 6 xmchanged, and the 7-head case has the follow- 

 ing sets of 3-triads: 



347 356 38 11 

 loading to a single triad system with 7-head but no triple tetrads. 



