262 Sir W. R. Hamilton on some 



tained in some of the preceding paragraphs, especially in [14.], 

 and as adapted to give some assistance towards any future study 

 of associative polynomes, such as quines, of an order higher than 

 quadrinomes, but subject like them to the law of conjugation (32). 

 The expression (98) may be thus more fully written : 



(f) = {f^ef + (feg){fye)+S{feh)(fke); . . (112) 



where efgh are all supposed to be unequal ; the summation ^ 

 being performed relatively to h, for all those w — 3 values of the 

 latter, which are distinct from each of the three former indices. 

 Interchanging e and ^, and subtracting, we eliminate the symbol 

 (/), and obtain the following formula : 



I- (feeY-(fggf=T{(fymhg)-(feh){fhe)}; (113) 



which type I. includes generally n[n—2) distinct and homo- 

 geneous equations, of the second dimension, with 2(w— 2) terms 

 in each, between the ^n^{n—\) symbols of the form (fgh). Thus, 

 for the case of quadrinomials (w=3), by writing, in agreement 

 with (82) and (83), 



«, = (!), ^ = (23), /i = (231), ;/z, = (313), 7.j = (122), (114) 



and suppressing the sum S\ we have by (112) the two expres- 

 sions (compare a note to [12.]) : 



a^-=m^—lj,^=n^^ — ljt^; .... (115) 

 together with four others formed from these, by cyclical permu- 

 tation of the indices 1, 2, 3 ; and we are thus conducted, by 

 elimination of the three symbols «,, a^, a^, to three equations of 

 the form n^^=m^'^; that is, to the 3 equations on the first line 

 of (86), involving each 2 terms. For quines {n=4), if we make 

 also, with the same permitted permutations, 



«, = (4), c, = (14), j».= (334), \ 



r, = (141), *, = (142), <,=(143), «, = (144),J 



the index h receives one value under each sign of summation S\ 

 and the resulting formulae may be thus written : 



(a^■\-lJ^-j-p^t^-s^p^=:^)n^^^ip^t^ = m^^s^P3=u^^-^lsIc,; (H^) 



where the line (117) is equivalent to three lines of the same 

 form : so that the elimination of Oi . . a^ conducts here to 8 equa- 

 tions, of 4 terms each, between the 24 symbols of the form {fyh), 

 or 1 1 . . Wg, as by the general theory it ought to do. For poly- 

 nomes of higher orders (n > 4), we have the analogous equations, 



(/) - iMifff") -(M) ifkg) - {fke) ifek) 



= (/««)' - ( /y*) (#y) + S ' (M)iM 



= (fgg?-(fkc)(fek)^-X\fgh){fkj) 

 =(fkkf-{fcg)(fge)^X\fkh)(fhk); . (119) 



