14 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol. xiv. 



Every pair al from one duad must occur in some one triad and can be associated with no third 

 element save A; hence seven triads are like Aal. The remaining 28 triads may be an-anged in 

 two pairs of classes, each pair equal in number by the symuietry of the operator S in letters 

 and digits, thus: 



X triads like abc, x triads like 123; 

 y triads like abl, y triads like al2. 



Since the nature of the system calls for 21 pairs of letters, as db, and 21 pairs of digits, while 

 there must be 42 mixed pairs such as a2, we have the equations of condition: 



3a; + i/ = 21,47/ = 42, 



insoluble in integers. Hence no triad system of this type can exist. 



§ 7. SUBSTITUTIONS OF TYPE (1)^(3)* AND THEIR INVARIANT TRIAD SYSTEMS. 



Denote by A, B, C three elements not affected by a certain substitution, and by (Oj a^ O3), 

 {h^ 6, 63), (Ci C2 C3), {di d^ d^, four cycles of period 3 in that substitution. 



S^{A) {B) (C) {a, a, a,) {h, I, \) (c, c, c,) (rZ, d, d,). 



Any triad system invariant imder S must contain the triad ABC; and equations of condition 

 show that of triads and sets like a^a^a,, b^h^c, there are respectively either 1, 3 or 4, 0. It 

 will be shown that this gives in letters, irrespective of their subscripts, five possible schedules. 

 There are 18 triads like Aah, falhng uito tliree sets of twice three. By the subscripts of these 

 latter each of the first five schedules is made a source of five subclasses. 



The equations of condition show further that there must be either two or four sets of thi-ee 

 triads like abc, formed from three different cycles in S. Hence there are at least three different 

 pairs of letters, as ab, that can not occur more than twice (i. e., 2X3 times) in conjunction 

 with letters A, B, or 0. The complementary pairs are excluded thereby also ; e. g., a set of triads 

 Aah would imply another set Acd, since all possible pairs of elements occur in a system A15. 

 Where four triads of period 1 occur, like afi^a^, and therefore four sets like abc, no pair ab 

 can occur with two letters from the thi-ee A, B, C. We arrive by such considerations at the 

 first five main divisions. 



Case 1. — Four triads like afi^a^, fom- sets like abc. Hence the schedule: 

 afi^a^, b^b^bs, c^c^c^, d^d^d^ 

 abc, abd, acd, bed; 

 Aab, Acd; Bac, Bbd; Cad, Cbc. 



Cases 2 and 3. — One triad of period 1, d^d^d^; two sets drawn from three different cycles. 

 Consider the triad sets containing pairs aa, bb, or cc. With these may occur the letter d in 

 3 2 1, or sets. If in none, then we must have (if lettera are chosen suitably) aab, bbc, cca. 

 But tliis leaves us to construct triads in which d^, for example, shall be miited with all nine 

 letters a, b, c. Three of these are of course m sets Aad, Bbd, Ccd, implymg sets Abc, Bac, Cab. 

 In the remaining two sets of three, d must be united twice with each letter a, b, and c, a plain 

 impossibihty. Similar absurdity results from assmning two sets like daa, dbb. Ilyisotheses of 

 one such set, or of three such, are admissible, as follows: 



Case 2: Assume two sets, a<id and bba. Trial of ccb leads to absurdity, whence we must 

 have cca. In full, therefore, this schedule is: 



ABC; d.d^d^; Abc, Aad, Bac, Bbd, Cab, Ccd; aad, bba, cca; led, bed. 



Case 3: Assimie thi-ee sets with d, aad, bbd, and ccd. The full schediile will be: 

 ABC; d^d^d^ Abc, Aad, Bac, Bbd, Cad, Cbc; aad, bbd, ccd; abc, abc. 



Cases 4 and 5.— With ABC and d.dj^ as in the preceding case, take dupUcate pahs of 

 letters with two of the isolated elements ABC; e. g., 



Aab, Acd, Bab, Bed; Cac, Cbd. 



There are yet to be constructed 15 ( = 5X3) triads. Of these five sets three have to contam a 

 doubled letter, as aa, while the other two consist of distmct letters. Listing the pahs that 



