PROFESSOR C. .T. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 295 
111. It is obvious from this theory that the united points of functions of this 
system compose definite tetrads, so that one point of a tetrad being given the 
remaining three are generally determinate. 
In fact (434) is a quartic transformation connecting united points q ■with the 
auxiliary points q), so that one point p corresponds to one point q, while four points q 
correspond to one point jx For a given point p), these four points are by (434) the 
intersections of the quartic surfaces, for arbitrary quateiiiions /, 
^i(v) = (t) /.X 
s/ii> S4p s/gp S4 p .V f 
But these surfaces have a common curve (435); and three surfaces having a 
common curve intersect in 
(xvp — m (/X + r + p — 2) + r.(145) 
points not on the common curve, and this number is 4 when p = = p = i, 
m = 10, r = 40, as in the present case. 
112 . The locus of points “ p ” determining functions, each of which has a united 
point on a given line, is a unicursal twisted (quartic. 
When we replace q by p + X(f in the second form of (434), we niay write 
P = ■==^ .(446), 
and the form of the equation establishes the proposition. 
In like manner we have 
t = (QhWg^Ia;, 1)^ = 4.(447). 
113. P'or every intersection of the line with the criticcd curve, the (quartic breaks uqr 
If X IS the value of the scalar x for a point on the critical curve, and tj,, both 
vanish, or 
^ = {p^dhPiPzPiJ.^', ^)\ U ^ (Vi¥34l^^''> 1)^'• • • • (448). 
We may employ these equations to eliminate p,^ and 4 from (446) and (447); and 
discarding the factor x — x', w’e find 
P = {p'i}P\Pip'‘ 6 X^^f^ 
(449). 
The locus of p is now a twisted cubic, and the discarded factoi’ corresponds to a line 
of the nature of those of Art. 1U9. 
When the line (qq meets the critical curve twice, the locus is a conic and a pair of 
lines. If the line is a triple chord, the locus is one line of a new tvpe and three lines 
of the type already mentioned. Finally, for a quadruple chord, the c|uartic reduces 
to a point and four lines, as we shall see immediately. 
But first we notice that the arguments of Art. 110 apply, so that we may write 
down the equation of tlie quartic curve whose jioints determine functions, each of 
