PROFESSOR C. J. JOLF ON QUATERNIONS AND PROJECTIVE GEOMETRY. 313 
These curves are sextics, and because (528) may he replaced hy 
[/" ( f" f” (p'h)] = 0, [/' (rp2), f ( tmi), f (rgc/)] = 0 . (530), 
vdiere [rp-p’g] = 0, they may be described as the locus of Jacobian correspondents 
of points in the plane reciprocal to the point r (424), 
As in Art. 109, when a point {q) is on the critical curve, its homograph is a line 
11.]' [q) - iG,^ (r) -f- FJp), Fj {r) = 0.(531), 
and not a point; and as in Art. 112 the homograph of a line + xq is a twisted 
cubic 
V = {PoPiV-GhJs^^f .(532); 
and a line of the type (531) breaks off the cubic for every intersection with tlie 
critical cui've. 
llius, when the line is a chord of the critical curve, its liomograph is also a line, 
so that 
{f{p + ^ r] = 0 .(533). 
Symmetry shows that p -|- xp must be a chord of the second critical curve. 
7/ the homograph of a line is plane, it is at most a conic. For the condition 
of planarity (compare Art. 120) 
{PoPiPiP-i) = ^ .(534) 
is of the sixth order in q and in q', and this equation represents a complex of the 
sixth order. But this complex can include nothing except intersectors of the 
critical sextic, for the cone of intersectors from the arbitrary point q is of the sixth 
order. 
The I’uled surface of triple chords has been noticed in Art. 75. 
145. The homograph of a plane 
S/q=0 is S/F^,(7-) = 0 
a general cubic surface through the critical curve. 
This cubic surface also passes through tlie sextic 
(535), 
.(53G), 
and it intersects the Jacobian I {p) — 0 in this sextic and in the critical curve. 
The equation of the Jacobian may be written in the forms 
(0 F, (0 = sy; k: (o = Hp) = o.(537), 
and for I and r, both variable, the curves Tq, (?■) = 0, FJ (1) = 0 generate tlie Jacobian 
in a manner analogous to the double generation of a quadric. Since the rank of the 
sextic is r = 16 (Art. 64), the two curves intersect in 14 points (477). 
VOL. cci. —A. 2 s 
