268 Sir WitiiAm Rowan Hamiton’s Researches respecting Quaternions. 
} ] 
mined by the condition that 7, or ¢,w, is, by (352), to receive the same sign as 
3 —f; but 7 is supposed positive, therefore w has the same sign as ¢; and 
wi+f)=wil +t) (356) 
so that the constituent w is entirely determined: therefore, 7 (being = tw) is 
known, and then the three remaining constituents, , y, z, of the sought quater- 
nion, g, are given by (350). Thus, the sought cube root, g, of the proposed 
numeral quaternion q,, is, in general, determined; or, at least, is restricted to a 
finite and ¢riple variety, answering to the three (real, numerical, and) wnequal 
roots of the known cubic equation (354) ; which roots can always be found by 
the help of a table of trigonometric tangents. We see, then, by the foregoing 
process, which will soon be replaced by one more simple and more powerful, that 
there are, in general, three, and only three, distinct cube roots of any proposed 
numeral quaternion. But when it is required to find, on the same plan, under 
the form of a quaternion, the cube root of a positive or negative number, w,,, 
regarded as an abridged expression for the quaternion (7,, 0, 0, 0), then 2,, 4 
z, and r,, all vanish; and while the ratios of z, y, z remain entirely arbitrary, 
the numbers w and r are to be determined so as to satisfy the two equations, 
Ws == 1 — Sur Ol 7.) (357) 
which require that we should suppose either 
P= Ose S78 (358) 
or else, 
r=3u, w= — tw, (359) 
For example, if we seek the quaternion cube roots of positive unity, regarded as 
equivalent to the quaternion (1, 0, 0, 0), we find not only unity itself, under the 
form of the same quaternion, but also this other, and partially indeterminate 
expression, 
1 10h, 0,10 ON ——i(— 2, a, ace) = (360) 
where the three positive or negative numbers, «x, 7, z, are only obliged to satisfy 
the condition 
4 Ppt? 2 (361) 
And, in like manner, besides negative unity itself, there are infinitely many qua- 
ternion cube roots of negative unity, included in the expression 
(—1)'= (—1, 0, 0, 0)'= (+42, y 2), (362) 
ee 
