128 Sir W. R. Hamilton on some 



equal to or to 1, according as e and k are unequal or equal to 

 each other. And no equations distinct from these are obtained 

 by supposing e=/7, or f=ff, in (56) and (59). The associative 

 conditions for which k = arc, therefore, in number, w(7i— 1) of 

 the form (60), and Jw(w— 1) (n — 2) of the form (56) ; or -^(n^ — w) 

 in all. And the other associative conditions, for which A: > 0, are, 

 in number, 7i*(7i— 1) of the form (61), and ^n^(/i~l)(n— 2) of 

 the form (59), or ^(w'*— n^) in all. It will, however, be found 

 that this last number admits of being diminished by ^(n^— w), 

 namely by one for each of the symbols of the form {fff) ; and 

 that if, before or after this reduction, the associative equations for 

 which X; > be satisfied, then those other ^{n^—n) conditions 

 lately mentioned, for which ^=0, arc satisfied also, as a neces- 

 sary consequence. The total number of the equations of associa- 

 tion, included in the formula (54), will thus come to be reduced to 



|(n4_n3)_^(n2-n), or to l7i(w-l)(w«-l) ; 

 but it may seem unlikely that even so large a number of condi- 

 tions as this can be satisfied generally , by the ^w(n^ + 1) constants 

 of multiplication [5.]. Yet I have found, not only for the case 

 71 = 2, in which we have thus 5 constants and 3 equations, but 

 also for the cases w = 3 and w = 4, for the former of which we 

 have 15 constants and 24 equations, while for the latter we have 

 34 constants and 90 equations, that all these associative condi- 

 tions can be satisfied : and even in such a manner as to leave 

 some degree of indetermination in the results, or some constants 

 of nmltiplication disposable. 



[1 0.] Without expressly introducing the symbols [fyh)^ results 

 essentially equivalent to the foregoing may be deduced in the 

 following way, with the help of the characteristics [3.] of opera- 

 tion, S, V, K. The formula of association (51) may first be 

 written thus* : 



tStV' + tV^V'=Sa'.^" + Vtt'.t"; . . . (62) 



in which the symbols Stt' and Vtt' are used to denote concisely, 

 without a point interposed, the scalar and vector parts of the 

 product u, but a point is inserted, after those symbols, and 

 before l", in the second member, as a mark of multiplication : so 

 that, in this abridged notation, Stt' . t" and Yli! . t" denote the 

 products which might be more fully expressed as (S . id) x t"and 

 (V . u') X l" ; while it has been thought unnecessary to write any 

 point in the first member, where the factor t occurs at the left 

 hand. Operating on (62) by S and V, we find the two following 

 equations of association, which are respectively of the scalar and 



* There is here a shght departure from the notation of the Lectures on 

 Quaternions, by the suppression of certain points, which circumstance in 

 the present connexion cannot produce ambiguity. 



