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IV. On the Triadic Arrangements of Seven and Fifteen 

 Things. By A. Cayley, Esq* 



THERE is no difficulty in forming with seven letters, 

 a, b, c, d, e,f, g, a system of seven triads containing 

 every possible duad; or, in other words, such that no two 

 triads of the system contains the same duad. One such system, 

 for instance, is 



abc, ade, afg, bdf, beg, cdg, cef; 



and this is obviously one of six different systems obtained by 

 permuting the letters a, b, c. We have therefore six different 

 systems containing the triad abc ; and there being the same 

 number of systems containing the triads abd, abe, abf and abg 

 respectively, there are in all thirty-five different systems, each 

 of them containing every possible duad. It is deserving of 

 notice, that it is impossible to arrange the thirty-five triads 

 formed with the seven letters into five systems, each of them 

 possessing the property in question. In fact, if this could be 

 done, the system just given might be taken for one of the 

 systems of seven triads. With this system we might (of the 

 systems of seven triads which contain the triad abd) combine 

 either the system 



abd, acg, aef, bee, bfg,dcf, deg, 

 or the system 



abd, acf, aeg, beg, bef, dee, dfg. 



But any one of the other abd systems would be found to 

 contain a triad in common with the given abc system, and 

 therefore cannot be made use of. For instance, the system 

 abd, acg, aef, bef, beg, dee, dfg contains the triad beg in com- 

 mon with the given abc system ; and whichever of the two 

 proper abd systems we select to combine with the given abc 

 system, it will be found that there is no abe system which does 

 not contain some triad in common, either with the abc system 

 or with the abd system. 



The order of the letters in a triad has been thus far dis- 

 regarded. There are some properties which depend upon 

 considering the triads obtained by cyclical permutations of the 

 three letters as identical, but distinct from the triads obtained 

 by a permutation of two letters or inversion. Thus abc, bca, 

 cab are to be considered as identical inter se, but distinct from 

 the triads acb, cba, bac, which are also identical inter se. I 

 write down the system (equivalent, as far as the mere combi- 

 nation of the letters is concerned, to the system at the com- 



* Communicated by the Author. 



