270 Sir Wirtam Rowan Hamitton’s Researches respecting Quaternions. 
we shall only have to change , in the expression for g, to 7r + jy -+ kz, and to 
suppose, as before, that 2° y* +2 =r’. But the process by which the two 
numbers w and r are thus supposed to be discovered, is precisely the process by 
which a numeral couple (w, 7), of the kind considered in the nineteenth article 
of this paper, and in the earlier Essay there referred to, would be determined, so 
as to satisfy the couple-equation, 
0=a,+a,(uv,7r) +a, (wr) + Ke. (368) 
The calculations required for finding a couple (w,r) which shall satisfy tsis 
equation (368), are therefore the same as those required for finding a quater on 
(w, x,y, 2), which shall satisfy the equation 
0= a, +4, (w, 2,4, 2) +a, (w, ay, 2)? + &e. ; (369) 
provided that we suppose the constituents of these two numeral sets to be con- 
nected with each other by the relation already assigned, namely, 
P+ytef=r. (336) 
Thus, in particular, if it be proposed to satisfy, by a quaternion g, the quadratic 
equation, 
O=a,+49749, (370) 
which we may put under the form 
g — 2aqg+b=0, (371) 
we may first change g to the couple (w,7), and so obtain the éwo separate equa- 
tions, 
w? —r—2au+b=0; 2wr—2ar=0; (372) 
of which the latter requires us to suppose, either, 
Sty 77 Os ory2nd,.70— a. (373) 
The first alternative conducts to a quadratic equation in w, namely, 
w? — 2aw +b=0, (374) 
which is precisely the proposed equation (371), with the symbol g of the sought 
quaternion changed to the symbol w of a sought number; and reciprocally if 
it be possible to find a real number w, or rather (in general) two such numbers, 
which shall satisfy the quadratic (374), that is to say, if (the equation have real 
roots, or if ) the condition 
a S10 0nd —10|=167, (375) 
