132 Sir W. R. Hamilton on some 



nation of the 6 constants of the forms here denoted by a and b, 

 or previously by (/) and {fy)j furnish 18 equations of condition 

 between the 9 other constants, of the forms here marked /, m, n, 

 or previously (fyh) ; and these 18 equations may be thus 

 arranged* : 



0=/9(»i-»»i)=4K-m2) = /i(n3-m8), I . . . . (85) 

 0=/3(ni-mO = /i(/i2-m2) = /2(n3-w?3) ;J 



0=(n2 + Wl2)(Wi— 7Wi) = (/l3+7W3)(W2 — W2) = K+'Wl)('*3 — '«3)> 



0=(w3 + W8)(Wi-m,) = (n, + m,)(n2-?W2) = K + W2)(w3-m3);J 



they are therefore satisfied, without any restriction on l^j^^^ by 

 our supposing 



w,=wii, 712=^2, ^3=7713; . . . (87) 



but if we do not adopt this supposition^ they require us to admit 

 this other system of equations, 



= /i=/2=^8=Wi + %='*2 + ^2='*3 + %- • • (88) 

 Whichever of these two suppositions, (87), (88), we adopt, there 

 results a corresponding system of values of the six recently eli- 

 minated constants, of the forms a and h, or (/) and {fg) ; and 

 it is found t that these values satisfy, without any new supposition 

 being required, the ^(/i^ — 7i)=8 scalar equations, included in 

 the general form 



S(t.tVO = S(tt'.^'0, .... (89) 

 which are required for the associative property. 



[13.] In this manner I have been led to the two following 

 systems of associative quadrinomials, which may be called systems 

 (A) and (B) ; both possessing all those general properties of the 

 polynomial expression P, which have been considered in the pre- 

 sent paper ; and one of them including the quaternions. 



For the system (A), the quadrinomial being still of the form 

 (81), or of the following equivalent form, 



Q=w + tx + Ky-\-7\^j (90) 



where wxyz are what were called in [1.] the constituents, the 



* For it is found that each of the three constants {eff)-\-{eg g) must 

 give a null product, when it is multiphed by any one of the constants 

 {e'fg'), or by any one of these other constants, {e"f'f")—{e"g"g") ; if 

 each of the three systems, efg, e'fg', e"f"g"y represent, in some order or 

 other, but not necessarily in one common order, the system of the three 

 unequal indices, 1, 2, 3. 



t This fact of calculation is explained by the general analysis of [15.]. 

 The values of a and b may be deduced from the formulae, ai=mi^— /j/a, 

 6,=/iwij— msng, with others cycUcally formed from these. 



