248 Sir Wizt1am Rowan Hamitton’s Researches respecting Quaternions. 
where i, j, & are still those three coordinate symbols, or new fourth roots of unity, 
already introduced in this Essay, of which the squares and products are subject to 
the fundamental formula : 
elim | Wel eer eal (A) 
and to the relations which are consequences of this formula, especially the fol- 
lowing: 
yo=—jak; gh=—W=t; R= —th=y. (B) 
These equations, (4) and (8), had indeed occurred before in this paper; but on 
account of their great importance in the present theory, they have been written 
once more in this place, in connexion with the general expression (c), which 
may represent any numeral quaternion. 
On the more general System of Coefficients, obtained by a recent Investigation. 
28. If we had not adopted the particular numerical values (241), but had 
allowed the four letters a, 6, c, d, in the equations (240), to denote any four 
constant numbers, which numbers, or their symbols, should thus enter as arbi- 
trary constants into the expressions for the coefficients of multiplication, and into 
those for the connected coefficients of derivation of quaternions ; then it is not 
difficult to see that, with the same fundamental system of expressions for the 
squares and products of 7, 7, k, contained in the formula (A), the results of the 
investigation in the twenty-fourth and twenty-fifth articles might be concisely 
presented as follows : 
Me Xo + m, xi +m, Xo =: Ms X3 == 
(m+ m,i+m,7j +m,k) (a+bi +e 4+ dk). (255) 
And then the formula of symbolic multiplication of one numeral quaternion by 
another, which is included in (152), namely, 
mi Xo- mi X, + my Xo +m; X,= 
(mi Xm K+ mM, Xp Ms X53) (1M, Ko + M, XK, + M, Xo+ MsX;), (256) 
would become, with the same system of non-linear relations between the same 
three symbols 7, 7, / : 
my +m t + mij + ms k= 
(ms, + mii + mij + mk) (a+ bi +o + dk) (m,+mi-+m,j+ m,k). (257) 
id ae 
