No. 2.] 



TRIAD SYSTEMS— WHITE, COLE, CITMMINGS. 



19 



By trials it is quickly proved that the third alternative gives a schedule which can not be com- 

 pleted to a full system. There remain then oidy cycles with two, four, or six pairs. Each cycle 

 can be divided into halves so that each half contains the same subscripts as the other (by taking 

 alternate pairs in the cycle) . This gives us twice three pairs involving all six subscripts. One 

 set of paii-s corresponding can be chosen of letters aa or hb, the other of unlike letters, ah. All 

 these possibilities, together with the self -conjugate triads, are outlined in the following; exhaust- 

 ing the possible schedules for triads in A, B, and C, except for nonessential substitutions. 



Class VI, 3 — Triads in A, B, and C, after ABC. 



Condensed tables nf subscripts. 



The cycles above described are seen in the second and third columns of these tallies (under 

 aa, hb and ab, ba). There are: One each of the first two species, two of the fourth. 



Altogether in the tlu-ee classes VI, 1, 2, and 3, we have therefore 11 schedules or partial 

 systems. These 11 can all be completed to full systems, most of them in two or more ways. 

 All the systems contain the triad ABO; the 18 triads containing a single A, B, or C are given 

 above; and the various supplementary sets of 16 triads, of the types aaa, bbh, aab, and ahb are 

 now to be listed in full. 



sxtpplementary sets, to complete the foregoing systems. 



(Of two conjugate triads only one is given.) 



VI, l,a. aaa: 135, 246. 



aab: 145, 164, 326, 361, 523, 542. 

 VI, l,a'. aaa: 135, 246. 



aab: 146, 163, 325, 362, 524, 541. 



Equivalent to VI, l,a by the substitution (16) (34) (25). 

 VI, l,/3. aaa: 135, 146. 



aab: 235, 246, 254, 263, 361, 451. 

 VI, l,/3'. aaa: 135, 146. 



aab: 236, 245, 253, 264, 361, 451. 



Equivalent to VI, 1,|8 by the substitution (36) (45). 

 VI, L. aaa: 145, 235. 



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



A second supplementary set is equivalent to this by the substitution 

 (12) (34). 

 VI, 13a. aaa: 135, 246. 



aab: 145, 162, 324, 361, 523, 546. 



An equivalent set is derived by the substitution (12) (36) (45). 



