74 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv. 



The substitution (123), with proper adjustment of the remaining numbers, converts the first of 

 these into the other tliree. The first may then be taken as typical. It has the following group 

 of six substitutions into itself: {(12 3) (4 8 10) (5 7 14) (6 13 11), (1 2) (11 13) (14 10) (9 15) 

 (5 8) (4 7)}. This suggests the formation of the triad 9 15 x. Here x can not be 1, 2, 3, 4, 5, 

 7, 8, 9, 10, 14, or 15, since 1 9, for example, has already been used in a triad; for % we must take 

 6, 11, 12, or 13. But 6, 11, and 13 are equivalent under the group of order 6 above. And the 

 remaining possibilities 9 15 6 and 9 15 12, if followed out, lead to tetrads or hexads. 



Returning to the table A, we may note that in the fir&t 13 cases the 3-triad which joins 

 a and h must be 3 5 7, for 3 4 6 is at once excluded, and 3 4 7 and 3 5 6 involve tetrads of 1 

 and 4 and of 2 and 5, respectively. Starting with 3 5 7, and WTiting in the remaining 3- 

 triads, we find that the first, second, sixth, and eighth cases lead directly to tetrads or hexads. 

 The remaining cases prove to be partly equivalent to each other, and those which survive are 

 found on continuation to the 4-triads, etc., to involve tetrads or hexads. 



3. THE HEXAD OPENING. 



This has the following group of 144 substitutions into itself: 

 {(4 7 8) (5 6 9), (4 5) (6 8) (7 9); 

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

 (4 10) (5 11) (6 12) (7 13) (8 14) (9 15); 

 (1 2) (5 6) (7 8) (11 12) (13 14)}. 



The triad 3 4 a; can not have x= 1, 2, 3, 4, 5, or 6, since these have already been used; nor can 

 x=7 or 8, smce these give tetrads of 4 and 1 or of 4 and 2. If the triad system is to have no 

 tetrads, we must take x = 9, 10, 11, 12, 13, 14, or 15, and of these the last six are equivalent 

 imder the group of order 144 above. Hence the only distinct tj^jes are 3 4 9 and 3 4 10. 



Starting, then, with 3 4 9, we find for 3 5 i/, only y = 7 or 10, and note that 3 5 10 is 

 equivalent under the group above to 3 4 10, the case to be considered later. It turns out, 

 then, that we can have only 



349 357 3 68 3 10 15 3 1113 3 12 14 



and we note that this is invariant under the group of order 144 above. If we now write in 

 the 4-triads, we come at once to tetrads. 



The case 3 4 10 leads to 3 9 11, 3 9 13, or 3 9 15. Following each of these out in detaU, 

 we encounter everjrwhere tetrads, with the single exception of the Ileffter system: 



This has a group of 768 substitutions: 



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

 (8 9) (10 11), (8 10) (9 11), (8 11) (9 10); 

 (12 13) (14 15), (12 14) (13 15), (12 15) (13 14); 

 (4 8 12) (5 9 13) (6 10 14) (7 11 15), 

 (4 8) (5 9) (6 10) (7 11); 

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



