264 Sir Wimuiam Rowan Hamitton’s Researches respecting Quaternions. 
thus, g' will denote that numeral set, or any one of those numeral sets, which 
satisfy, or are roots of, the equation, 
Vy=e=e (326) 
For example, it results from what has been already shown, that if g denote the 
first numeral quaternion (2959), then its symbolic square, or second power, is 
another quaternion, ¢,, given by the formula 
=F =(w+ix + Jy + kz)? = w, + 12, + IY> + hz2, (327) 
2 
— a2 a2 2 m2 e 
wv, =u —xr—y— 2; 
where ; 
Li 2WE3) Ya — 2WY ie eee j ney 
And hence, conversely, the symbolic sguare root of the quaternion q,, or its 
power with the exponent $, is to be regarded as being equivalent to this other 
numeral quaternion, 
q = 4e = (w, + tx, + jy. + kz,)* = w + te + jy + kz; (329) 
where the constituents, w, x, y, z, are any four numbers (positive, negative, or 
zero), which satisfy the system of the four equations (328). Those equations 
give the relation 
w+ 4y+y +2 =W+ew’tyt a), (330) 
which is included in the more general result (251), respecting the multiplica- 
tion of any two quaternions ; therefore, conversely, 
wp at hyp t= VM (we fame + yet +23); (331) 
and, consequently, by the first of the four equations (328), 
2w = w,4+V(wP+ae+ ye + z,), (332) 
where the radical in the second member of (331) is to be considered as a positive 
number: and, therefore, the first constituent, w, of the sought quaternion g, or 
of the square root of the given quaternion g,, is itself given, generally, by (332), 
as either the positive or the negative square root of another given positive num- 
ber. And after choosing either of these two values (the positive or the nega- 
tive) for w, the other three constituents, x, y, 2, of the sought quaternion q, 
become, in general, entirely determined by the three last equations (328). 
There are, therefore, i general, two, and only two, different square roots of 
any proposed numeral quaternion ; and they differ only in their signs. But there 
a 
