Sir Witt1Am Rowan Hamirton’s Researches respecting Quaternions. 265 
is one very important CASE OF INDETERMINATENESS, in which an 7finite variety 
of roots takes the place of that finite ambiguity, which has thus been seen to 
exist generally in the expression for the square root of a quaternion, namely, 
the case where the proposed square is equal to a negative number, presented 
under the form of a quaternion, of which the first constituent is negative, while 
the three last separately vanish. For, if we suppose the data to be such that 
i= — G77, 5 =0,- 45= 0, 25 = 0, (333 ) 
r being some positive or negative number, then the positive radical in (331) 
becomes 
V(we+ ae + ye +27) =r=— wy (334) 
and the equation (332) reduces itself to the following : 
uw 0: (335) 
And while the three last of the four equations (328) are then satisfied, indepen- 
dently of the three remaining constituents, x, y, 2, the first of those four equa- 
tions gives this ove relation, between those three constituents of the sought qua- 
ternion q, 
eY+y+2=r, (336) 
which is the only condition that they must satisfy. And since we may satisfy 
this condition by assuming 
SSN es 
War Vilitiate Pe Gn enables (337) 
h=V(P+m'+n’), 
without any restriction being imposed on the three (positive, or negative, or null) 
numbers, /, m, ”, we see that, in our theory of quaternions, the sguare root of a 
negative number is a partially indeterminate quaternion, belonging, however, 
to a certain peculiar c/ass, and admitting of being thus denoted : 
re (a +- ym + kn)r 
( ™) a: JV (P+ m+ n*) (338) 
In fact, if we square the second member of this last formula, attending to the 
fundamental expressions, (A), (B), for the squares and products of the three 
symbols, 7, 7, &, we find, as the result of this operation, the negative number 
2Nn2 
