Sir Wizi1AM Rowan Hamirton’s Researches respecting Quaternions. 267 
and this ewbe, or third power of a quaternion, may be equated to a new quater- 
nion, denoted as follows : 
=P =U, t iv, + jy, + kz, = 0,4 ors (347) 
fs ae-pye es, Ss (348 ) 
where 
provided that we satisfy the two conditions, 
w, = w* — 3ur’, u7r,= r(3u —7’), (349 ) 
of which the second again resolves itself into three others, on account of the 
mutual linear independence of the three symbols, 7,7, &. These last equations 
give 
3 Is — 3 — gy _ (350) 
a eae 
and, therefore, it is allowed to write 
habs fy 1(3w* — 7) 5 (351) 
provided that, if we still choose to consider the radical 7 as positive, we regard 
the other radical, 7,, as varying its sign, according to the law 
r, 2 0, according as 31? 2 r°. (352) 
If, now, it be required to find conversely the cube root q, or the power with 
exponent + of a given quaternion, ¢,, we shall have, first, the two equations 
of which the second may be written more concisely thus : 
3¢ —fP = (1 — 3? )t, ifr ste, 17, = t,0,; (354) 
so that 
ty =w, *(@; +43 + 2;')- (355) 
The value of this positive number, ¢,’, is known, because the four constituents of 
the quaternion g, are now supposed to be given; hence, three different positive 
values for ¢ can, in general, be deduced from the square of the first equation 
(354), which is a well-known cubic; for each such value of ¢*, the sign of ¢,, 
and therefore, also (by the same cubic equation), the sign of ¢ may be deter- 
