On the Triadic Arrangements of Seven and Fifteen Things, 51 

 mencement of this paper) 



ade, afg 9 bdf bge, cdg, cef 9 cba, 



derived, it is to be observed, from a pair of triads, ade, afg 9 

 by a cyclical permutation of the e 9 f 9 g 9 and by successively 

 changing the a into b and c 9 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 9 ade 9 gcd 9 geb 9 fce,fbd 9 fga* 9 



which is, in fact, the original system. But attempt to derive 

 the system from the two triads ebg 9 efc 9 the process of deriva- 

 tion gives 



ebg, efc 9 dbf, dcg 9 abc, agf 9 ade 9 



which is not the original system, inasmuch as the triads dbf 

 dcg, abc 9 agfare inversions of the triads bdf ) cdg 9 cba, afg of 

 the original system. The point to be attended to, however, 

 is, that both triads of the pair dbf, dcg 9 or of the pair abc, agf 9 

 are inversions of the triads of the corresponding pair in the 

 original system ; the pair is either reproduced (as the pair efc 9 

 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 9 c or d, and improperly reproduced from 

 pairs of triads containing in common any one of the letters d 9 

 e or/. 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 circum- 

 stance 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 com- 

 bination of the letters (viz. that aj3y and ads being any two 

 triads having a letter in common, there shall be triads such 



* The order of the letters/, g is selected so as to reproduce the original 

 system so far as the mere combination cf the letters is concerned. 



E2 



