PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 251 
Tait first extracted the square root of a linear vector function, and pointed out the 
great utility of the concejjtion. We now proceed to examine some of the properties 
of a square root of a quaternion function, and to illustrate tlmir bearing on certain 
geometrical investigations. 
llie united points of a seiuare root are also united fooints of the 'primitive function. 
If 
f^a = t^a, then fa — tpa , . . .(150). 
The converse does not hold, for it may hajD^^en that loci of united points exist for the 
primitive and not for the square root. For example, if 
f^Oj — b, f-h = tpa ; then fa = tpi, fh := tf> . . . . (151), 
and though every point on the line ah is, a united point for the primitive, this is not 
generally true lor its square roots. (Compare Art. 13.) 
When there is no locus of united points, the square roots have the same four united 
points as the primitive, and their latent roots are sets of the square roots 
i if i Ch i if i .(1^2) 
of the latent roots of the primitive. Thus in the general case a function has sixteen 
square roots. 
37. When the primitive has a line locus of united p)oints [a, b), any tivo points on 
the line may be assumed as united points of the square 'root. 
By the last article it may. be seen that the square root must have united points 
on the line. Assume these to be a + xh, a + y/>, then 
/I (a f-xb) — ± qi(a + xb) ; p {a yf = ± id (« W V^) • • (1^3), 
and the square root satisfies the condition that its twice repeated operation is 
equivalent to the operation of f. If the signs are alike and x and y distinct, the 
square root has a locus of united points; otherwise it has not. 
If a square root has coalesced points, so has the p)rimitive. 
If 
Pa = thi + a ; f^-ct = lla .; then fd = td + 2^k^; fa = tct. . (154), 
and therefore the repeated operation of jf — t is required to destroy a ; and the 
primitive has a coalesced united point. 
The square root of a f unction having a plane locus of united points possesses at least 
Oj line locus of united points. 
The only escape is the assumption that the square root has a united point coalesced 
from three points, and this lias just lieen shown to involve a coalesced jioint for the 
primitive, contrary to hypothesis. 
When the pjrimitive has coalesced points but no loci of united qjoints, the number of 
square roots is limited. 
This follows from (154). 
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