NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 11 



Family la. 



The only possible schedule, prior to tlie assignment of subscripts, is this: 

 aaa, hhh; ace, add, ace; 

 abc, abd, abe; 

 bed, bee, bde; 

 cde, cde. 

 For the hypothesis ace, add, and bee, for example, would require a, to unite with two e's in 

 the one triad left for it after the three where it occurs with b's; whereas ee is found already 

 with b in bee, whence the impossibility. 



With entire generality we fix, in the above schedule, the subscripts in eight typical triads: 

 a, a., a^; &, h^ b^; Oi c, C3, a^ d^ d^, a, Cj €3] 

 (?! &i Ci, ttj 62 di> «i ^3 ^v 



Six trials now sufBce, to reduce the possible ways of supplying subscripts for the remaming 

 triads to the following two, supplementary to the above: 



System II, Ij: c, di b^, c, fZ, e^, c^ d^ e^; 



6, C3 fj, &i di Ci. 



System II, 1,: c, d^ 6,, c'l <?, (!„ c^ d^ e^; 



h <^2 «1) ^1 ^^2 «3- 



Family lb. 



Prior to assigning subscripts, the only possible schedule is this: 

 aaa, bbb; ace, add; cee; cde; 

 abe, abe, abc; 

 bed, bed; ode, bde. 



(The other apparent possibihty, aec, bdd, cee, leads to the impossibility of five times three 

 pairs ce.) 



Subscripts may be fLxed arbitrarily, in order, in the following triads (subject, of course, 

 to the cyclic permutation of 1, 2, 3): 



a^a^a^, bfij)^; afi^Ci, afi^e^^, afi^Ci, 

 aiC^d^; OiC^Cy, Qid^d^. 



The remaining five sets arc foimd by trial to admit two arrangements only. With the above 

 we may unite either of these: 



System II, 1,: biC^d^, System II, 1,: b^e^d^, 



^i<:A> b^c,d„ 



Cid^Cs, c^djCi, 



Family Ic. 



Tlio association of letters in triads, disregarding subscripts, may foUow two schedules. 

 The first possible schedule is this: 



aaa, bbb; 



ace, cdd, cee; 



abc, abd, abe; 



ode, ode, bde, bed, bee. 



In the first six that follow aaa and bbb we may dispose of subscripts by fixing a,5if,, then fljCCs, 

 c.rfjt/j, 0,^2^3. So far letters d and e are exchangeable, also subscripts 2 and 3. This observation 

 reduces to five the number of essentially different ways of affixing subscripts to the next two 

 triads, abd and aic, viz: 



(1) a^dfi^, (2) a,dfi„ (3) a,dj)„ (4) a.d.b^, (5) a,d,\, 

 0,6,63; o-ye^h; aiCz&a; O1C362; ciie^by 



