130 Sir W. R. Hamilton on some 



+ y(w^— 7i) associative and scalar equations of condition, between 

 the A(n^H-7i) disposable constants of multiplication, when the ge- 

 neral conception of the polynomial expression P of [1 .] is modified 

 by the suppositions, Iq=1 in [2.], and iU=Ki? in [5.]. At 

 least the analysis of the present paragraph [10.] confirms what 

 has been lately proved in [9.], that the number of the conditions 

 of association can be reduced so far; but the same analysis will 

 also admit of being soon applied, so as to assist in proving the 

 existence of those additional and general reductions which have 

 been lately mentioned without proof, and which depress the 

 number of conditions to be satisfied to ^{n'^ — n^) —^{n^—n)» 

 Meanwhile it may be useful to exemplify briefly the foregoing 

 general reasonings for the cases ri=2, /i=3, that is, for trinomial 

 and quadrinomial polynomes. 



[11.] For the case w=2, the two distinct symbols of the form 

 I may be denoted simply by i and i/ ; and the equations of asso- 

 ciation to be satisfied are all included in these two, 



t.a'=AV, c',L'c=c'h; .... (70) 



which give, when we operate on them by S and V, two scalar 

 equations of the form (68), and two vector equations of the form 

 (69), equivalent on the whole to six scalar equations of condition, 

 between the five constants of multiplication, (1) (2) (12) (121) 

 (122), if we write, on the plan of preceding articles, 



.^=(1), .'^=(2), Sa'=(12),n 



From (68), or from (60), or in so easy a case by more direct and 

 less general considerations, we find that the comparison of the 

 scalar parts of the products (70) conducts to the two equations, 



0= (121)(1) + (122) (12) = (122) (2) + (121) (12). . (72) 



From (69), or (61), we find that the comparison of the vector 

 parts of the same products (70) gives immediately four scalar 

 equations, which however are seen to reduce themselves to the 

 three following : 



(121)(123)=-(12); (122)5= (1); (131)«=(2); . (73) 



the fii'st of these occurring twice. And it is clear that the equa- 

 tions (72) are satisfied, as soon as we assign to (1) (2) and (12) 

 the values given by (73). If then we write, for conciseness, 



(121) = «, (122) = 5, (74) 



we shall have, for the present case (n = 2), the values, 



(1)=6», (2) = a«, (12) = -fl6. . . (75) 



