266 Sir Wrrr1am Rowan Hamitton’s Researches respecting Quaternions. 
—r*, which is the square of the first member; for those fundamental expres- 
sions give, generally, this very simple and remarkable equation, 
(ie + yy + kz =— (2° + y* +2’). (339) = (v) 
For example, in this theory, the square root of — 1 itself is represented by a 
partially indeterminate symbol of the foregoing class, and we may write 
(-Ii= S442 nes where 7? = a®° + 7? + 2”. (340) 
That is to say, whatever three positive, or negative, or null numbers may be 
denoted by x, y, z, provided that they do not all together vanish, we are allowed 
in this theory to establish the following general expression for any one of the 
infinitely many square roots of negative unity : 
(ae Fiat antes ; (341) = (e) 
ory +2) 
Or, with the recent meaning of 7, and with a notation which more immediately 
suggests the conception of a numeral set, we may establish the formula, 
(— 1, 0, 0, 0)'= (0, -, Z =), (342) 
Cubes and Cube Roots of Quaternions ; partially indeterminate Expressions 
by Quaternions for Cube Roots of positive and negative Numbers. 
37. With the same condition or abridgment, (336), we may write generally, 
for any numeral quaternion, this expression 
g=w+(-1)r; (343) 
or still more briefly and, at the same time, more determinately, 
g=w-+u, where °= — 1, (344) 
and where « may be conceived to be in general determined when q is determined, 
since 
i, I 
Stee, r=V(’+y4+). (345) 
The cube of this expression (344) for q is 
g=w — sur +.(38w — 7 )r; (346) 
— a. 
