No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 9 



System 12 : a.al, aby, ac4, a^c, a72, 125, 13(Z, and their conjugates under the operator S^. 



(2) Triad a74. 



By similar scrutiny it is found that triad a74 excludes the pair 1 2 from completion by the 

 Greek cycle, but allows the requisite triad 135 and its conjugates. To be tested are now 12c, 

 12(/, and \2e. Of these, 12c leads to ac5 and so to cc2 inconsistent with 12c; and the thirtl 

 alternative 12e woidd exclude the pair oc from the Arabic cj^cle altogether, and is hence inad- 

 missible. There remains I2d, which allows further ac2. Complete the system thus, uniquely, 



aa\, aby, ac2, 12d, 135, afic, ayi. 



This system difTors from 12, above, only in the interchange of the Greek cycle and the Enghsh. 

 Omit it therefore as a duphcate. 



(3) Triad ay 5. 



By considerations like the above we find that ay5 and aal with their conjugates will exclude 

 12 from completion in the Greek cycle, and wiU necessitate the occurrence of 13e, which in 

 turn requires 075. Next, 12 must be joined with either c, d, or e. The fii-st and last of these 

 lead to inconsistent triads, and we have remaining 12fZ, and therefore ac2. The sole admissible 

 system here is therefore that given by these seven: 



aal, aby, ac2, ay5, a/3c, 13e, 12c?. 



But this triad system is related to the operator S^, 



S{'={a yeP5)(lS524)(acehd), 



in exactly the same way as system 12, above, is related to the operator S^, 



S^=(a 6 c (Z c) (a /3 7 5 e) (1 2 3 4 5), 



We may therefore omit it as a duplicate. 



This exhausts the possibihties under the fii-st assumption, i. e., that the second triad was 

 aby. Test now the other alternative: Assume abS. By trials similar to the foregoing, we con- 

 struct the following five sj-stems, exhausting the possibihties: 



(a) aal, abS, ayb, ac4, o|83, I3d, I27. 



(b) fflal, abS, ayb, ac5, a/33, 13c, I27. 



(c) aal, abd, ayb, ac4, a/35, 13d, 12€. 



(d) aal, abS, ayb, ac5, afi5, 13c, 12e. 

 System 13: aal, abS, ayb, ac2, ajS-i, 13/3, 12d. 



Four of these systems are equivalent to the two already found. Notice first that two of 

 them, (d) and (c), reduce to (a) and (b), respectively, by the reversal of order in each cycle, i.e., 

 by using for operator S^* in place of 5",. Then (a) is seen to become system 12 by exchange of 

 English and Arabic cycles. To show that (b) is congi'uent to S3'stem 12, replace operator (Sj 

 by fS'i' in a changed order of cycles, thus 



S, =(a & c d e) (a j3 7 5 e) (1 2 3 4 5) 

 S','=(a 5/3 6 7) (14 2 5 3) (adbec) 



The same substitution wdl change (b) into 12. 



The net result of this section is therefore the construction of three systems, II, 12, and 13. 

 The fu-st and third of these are the well-kno^vn systems of Heffter and Kirkman, respectively, the 

 second hitherto unknown. 



