Sir Witt1Am Rowan Hamutton’s Researches respecting Quaternions. 269 
under the same condition (361) respecting the sum of the squares of the consti- 
tuents, x,y, z- The values of this last expression (362), as well as the values of 
the expression (360), are, therefore, included among those quaternions which 
are (in this theory) siath roots of unity, or are among the values of the symbol 1’. 
As one other example, it may be remarked that, by the rule (859), the number 
negative eight has, for one of its cube roots, the quaternion of which each of 
the four constituents is equal to positive unity; thus, one value of the symbol 
(—8, 0, 0, 0), is (1, 1, 1, 1); (363) 
and, accordingly, we shall find that 
(Ltitj+ey=—s, (364) 
if we develope the first member of this last equation, employing the distributive 
property of multiplication, but mot the commutative property, and reducing by 
the values of the symbolic squares and products of 7, 7, , which have been already 
assigned. It may be noted here that, in the more general problem of finding the 
cube root, g, of a quaternion, g,, of which the three last constituents, 2,, y., Z,, 
do not all vanish, so that r, is different from 0, we might have eliminated r* 
between the first equation (349) and the first equation (853), and so have 
obtained an ordinary cubic equation in 7, which, as well as the equation in ¢, can 
be resolved by the trigonometrical tables, namely, the cubic : 
dw? — 3w(w, +77) = wv, (365) 
Connexion of Quaternions with Couples, and with Quadratic Equations. 
38. In general, if a numeral quaternion g be required to satisfy any ordinary 
numerical equation (with real coefficients) of the form 
O=a,+ag+ a7 + ag + ke, (366) 
we may first substitute for q the expression (344), namely, w + u, where 
¢ = —1. Then, after finding any one of those systems of values of the two (real) 
numbers w and 7, which satisfy the system of the two equations, obtained by the 
foregoing substitution, and by equating separately to zero the sums of the terms 
containing respectively the even and odd powers of 4, namely, the equations 
0 =a, + aw +a,(u’—r’) + a,(w’—dur’) + &e., ] (367) 
~ v 
(= ar +a,2wr) + a,(3wr—r’) +&e.; | 
