276 Sir Witt1am Rowan Hamitton’s Researches respecting Quaternions. 
of x, while the direction of its vector unit is indeterminate or unknown, the for- 
mula (412) for a square root becomes simply 
(— r= Ph te 5 (413) 
the position of the point rR upon the wnt sphere being now likewise indeterminate 
or unknown, which agrees with our former results respecting the indeterminate 
quaternion forms for the square roots of negative numbers. In like manner, the 
quaternions, distinct from unity itself, which are cube roots of wnity, are now 
included in the expression 
1 = cos “2 + i, sin “2; (414) 
where the direction of 7, remaims entirely undetermined. But, in general, the 
e . . . . . . 
power, q», of a quaternion, q, admits of s, and only s, distinct quaternion values, 
2 t : ; Pe ta: 
if the exponent, ra be an arithmetical fraction in its lowest terms, so that the 
numerator and the denominator of this fractional exponent are whole numbers 
prime to each other; and if the proposed quaternion g do not reduce itself to a 
number w, by the three last constituents, 2, 7, 2, all separately vanishing in its 
expression. As an example of the operation of raising a quaternion to a 
power of which the exponent is distinct from all positive and negative numbers, 
and from zero, we may remark that the formula (410) gives, generally, for the 
powers of an imaginary unit, such as 7, (for which we have » = 1, @= 3) the 
expression 
ee ee v}qi,(5 +2nn)} 2 (415) 
making then, in particular, 7, = 72, and g’ = 4, we find, by (8), 
=v} hi(5 + 2nn)t =r}i(§+2nn)t = (4) ai: (416) 
and by a similar process we find, more generally, 
i.” = (417) 
whenever 7,, and 7, denote two rectangular imaginary units, so that the points 
r and r’, which mark their directions, are distant from each other by a quadrant 
on the sphere. We may here introduce a few slight additions to the nomencla- 
