NO. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 21 



VI, Sj. aaa: 145, 235. 



aab: 135, 162, 245, 263, 364, 461 ; 

 or 135,164,245,261,362,463. 



These two alternatives are equivalent by the substitution (12) (34). 

 We shall refer to the first. 

 VI, Sja. aaa: 146, 235. 



aab: 125, 136, 241, 362, 453, 564. 

 VI, 330. aa^: 146, 235. 



aab: 126, 132, 245, 364, 451, 563. 



Equivalent to VI, Sga by the substitution (12) (34) (56). 

 VI, 33T. aaa: 124, 136. 



aab: 251, 352, 453, 654, 236, 461. 

 VI, SjS. aaa: 412, 465. 



aab: 316, 325, 354, 362, 164, 251. 

 VI, S^a. aaa: 146, 235. 



aah: 124, 132, 436, 453, 625, 651. 

 VI, 34/3. aaa: 146, 235. 



aab: 125, 134, 432, 456, 621, 653. 



Equivalent to VI, 3,a by the substitution {BC) (12) (36) (45). 

 VI, 3^7. aaa: 126, 134. 



aah: 234, 251, 461, 352, 456, 563. 

 VI, 346. aaa: 312, 345. 



aab: 142, 253, 614, 625, 643, 651. 



Equivalent to VI, 3^7 by the substitution {AB) (13) (24) (56). 



In the above enumeration some systems can still be omitted as redundant. 

 No. VI, Sit' is reduced to VI, \^a by the substitution iflp^ (bfi^- 

 Five systems are reducible to VI, 2ia, viz: 



VI, 2i7 by the substitution {BC) {a^a^) (b^ag) (a^b^) (b^b^) (ajSJ; 



VI, 3, a by the substitution (ACB) {a^a^a^ (b.b^aj {a,_aJ)J)J)J)^): 

 and the tlu-ee whose equivalence to VI, 3ia has been noted akeady. 

 Further, No. VI, 3,j3' is reducible to VI, 2i5 by the substitution 



{B(7) (a^a^) {a J) J),) (a^bjy.ajt^b^). 



Some of those equivalences are obvious on comparison of the structure as here described, 

 but othere would not have been found without the aid of some definite system of procedure. 

 The method actually used was Miss Cummings's method of sequences and indices. 



2\iter these deductions for equivalence, there remam 21 systems apparently distinct, 

 automorphic under a substitution of the type (1)'(2)^ 



In wTiting down these supplementary sets of triads, the first step was to T^nite the two 

 required triads aia^a^ in all ways that are different as regards the schedide of triads in A, B, 

 and C; that is, in all possible ways not transformable into one another without alteration of 

 the preceding 19 triads of the proposed system. After each way of writing these two triads 

 OiOjak, it is easy to decide from mspection whether the number of ways of filling out the six 

 triads aab is 0, 1, or 2. Where possible pairs of triads a^afl^^ have been omitted, it indicates 

 the impossibility of filling out a system. 



§9. THE SUBSTITUTION OF THE TYPE (1)"(2)'': INVARIANT TRIAD SYSTEMS. 



The only remaining (reduced) type of substitutions is that which leaves unchanged 7 of 

 the 15 elements and exchanges the others in pairs. Denote the former by numerals or digits 

 1, 2, 3, 4, 5, 6, 7; the latter by the pairs of letters Aa, Bb, Cc, Dd. The operation to be con- 

 sidered is S : 



5=(l)(2)(3)(4)(5)(6)(7)(^a)(56)((7c)(Z?(i). 



