268 On some Extensions of Quaternions. '\ ' 



where e/ffh are again unequal indices. This IV.th type includes 

 generally n(n— l)(7i— 2) distinct equations, with n— 1 terms 

 each. For the case 7i=3 there arise thus 6 equations of 2 terms, 

 namely the six on the two last lines of (85) ; so that the 18 

 equations (85), (86), for associative quadrinomials, have thus 

 been completely reproduced, as consequences of the general 

 theory. For the case of quines, the type (151) gives 24 equa- 

 tions of 3 terms, which may be represented as follows : 



0=s^(r^-^r^) + t^t^ = ts{r^-\-r^)-^s^^ 





(152) 



= 5i(Mi+/Ii)— /2^1=;)3(Mi + Wj) — /3jt?2; ^ 



either A or e being =4 in (152), and either/ or y having that 

 value in (153), while 1, 2, 3 may still be cyclically permuted. 



[24.] Finally, by supposing, as in (119), that efgk are four 

 unequal indices, and that h under S^^ is unequal to each of them, 

 we obtain from (96) one other type, including ^^7iera% ^n[n — \) 

 (n— 2)(w— 3) equations, of 71— 1 terms each, but furnishing no 

 new conditions of association for quadi'inomials : namely, 



V. efk.gee+fgk . egg^-egk .fkk=:T\fhk .geh). (154) 



For quines, the sum S^^ vanishes, and we obtain twelve equations 

 of three terms each, which may (with the help of permutations) 

 be all represented by the foui' following : 



0=l^r^+n^9^-\-m^tci = nip^-m^p^-UsP^',S 



where the index 4 has been made to coincide with e or with g in 

 the first line, but with /or k in the second. 



[25.] In general, the number of distinct associative equa- 

 tions, included in the three last types (147) (151) (154), or 

 III. IV. v., which have been all derived from the formula (96), 

 and have been obtained without elimination of (/) or {fg), 

 amounts in the aggregate to 



Jn(/i-l)(n-2)+7i(n-l)(w~2)+Mw-l)(n-2)(?i- 3) 



=^n\n^l)(n-2); (156) 



as, by the analysis of [14.], it ought to do. And when we add 

 this number to the 7i(w— 2) of the type I., or (113), and to the 

 7i(n— l)(n— 2) of the type II., or (123), obtained by such ehmi- 

 nation, we have in all this other number, 



^n^{n-l){n-2)-\-n^{n-2)='^n%n + l){n-2), . (157) 



of distinct and homogeneous equations of the second dimension, 

 between the ^n*(7i— 1) symbols of the form {fgh) : as, by the 



