482 ON THE TRIADIC ARRANGEMENTS OF SEVEN AND FIFTEEN THINGS. [82 
The order of the letters in a triad has been thus far disregarded. There are 
some properties which depend upon considering the triads obtained by cyclical per 
mutations 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 combination of the 
letters is concerned, to the system at the commencement of this paper) 
ade, afg, bdf, bge, cdg, cef, cba, 
derived, it is to be observed, from a pair of triads, ade, afg, by a cyclical permutation of 
the e, f g, and by successively changing the a into b and into c, the remaining triad 
of the system being the letters a, b, c taken in an inverse order. Let it be proposed 
to derive the system in the same manner from any other two triads of the system; 
for instance, from the triads acb, ade. The process of derivation gives 
acb, ade, gcd, geb, fee, fbd, fga, ( x ) 
which is, in fact, the original system. But attempt to derive the system from the 
two triads ebg, efc, the process of derivation gives 
ebg, efc, dbf deg, abc, agf ade, 
which is not the original system, inasmuch as the triads dbf deg, abc, agf are in 
versions of the triads bdf, cdg, cba, afg of the original system. The point to be 
attended to, however, is, that both triads of the pair dbf deg, or of the pair abc, 
agf are inversions of the triads of the corresponding pair in the original system; 
the pair is either reproduced (as the pair efc, dbf), or there is an inversion of both 
triads. Where there is no such inversion of the triads of a pair, the system may 
be said to be properly reproduced; and where there is inversion of the triads of one 
or more pairs, to be improperly reproduced. There is no difficulty in seeing that the 
system is properly reproduced from a pair of triads containing in common any one 
of the letters a, b, c or d, and improperly reproduced from pairs of triads containing 
in common any one of the letters d, e or f It is owing to the reproduction, proper 
or improper, of the system from any pair of duads that it is possible to form a 
system of “ octaves ” analogous to the quaternions of Sir William Hamilton; the 
impossibility of a corresponding system of fifteen imaginary quantities arises from the 
circumstance of there being always, in whatever manner the system of triads is 
formed, an inversion of a single triad of some one or more pairs of triads containing 
a letter in common. When the system is considered as successively derived from 
different pairs,, the system is not according to the previous definition reproduced either 
properly or improperly. A system of triads having the necessary properties with 
respect to the mere combination of the letters (viz. that a/3y and aSe being any two 
1 The order of the letters /, g is selected so as to reproduce the original system so far as the mere 
combination of the letters is concerned.