82] 
481 
82. 
ON THE TRIADIC ARRANGEMENTS OF SEVEN AND FIFTEEN 
THINGS. 
[From the Philosophical Magazine, vol. xxxvn. (1850), pp. 50—53.] 
There is no difficulty in forming with seven letters, a, h, 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 contain the same duad. One such system, for instance, is 
ahc, 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 
or the system 
abd, 
acg, 
aef, 
bee, 
¥9’ 
def, 
deg, 
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, bcf, beg, dee, dfg contains the triad beg in common 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. 
C. 
61