PROFESSOR C. J. JOEY ON QUATERNIONS AND PROJECTIVE GEOMETRY. 
290 
The twisted quartics of Art. 112 correspond to the conics on the Steiner qnartic. 
The sextic surface contains ten lines corresponding to the ten points in which the 
plane intersects the critical curve of the tenth order, for to every point on that 
curve corresponds a two-system of functions or a line in the space (Art. 109). 
Again, the sextic contains an infinite number of twisted cubics corresjionding to the 
lines in the plane which pass through one of these ten points (Art. 113); and it 
likewise contains 45 conics answering to the connectors of these points. More 
generally (Art. 113) a conic through five of these points transforms into a twisted 
cubic, and similarly for other cases. 
120 . When we express that the twisted cubic (449) is plane, the condition 
= h.(473) 
is of the tenth order in q' and of the sixth in q, which latter point we may suppose 
to be on the critical curve. This condition will then represent a cone of the tenth 
order of the lines through the point q which transform into plane curves in the 
2 ^ space. But this cone must consist in part of the cone of the ninth order containing 
the critical curve. The remaining part is a plane, and every line in this plane 
through q transforms into a plane ciilhc. 
In paiticular, an arbiti’ary plane cuts the critical curve m ten points and intersects 
ten planes of the type just mentioned in lines which transform into plane cubics on 
the sextic surface. Here again is a jioint of similarity with the Steiner qnartic, for 
the plane containing one of these cubics cuts the sextic again in another cubic. 
121. Corresponding to a plane [_ 2 \ 2 '> 22 ^^ in the p space there is a Jacobian quartic 
filMl /(M)) = b.(474) 
111 the q space, the locus of united points of functions of the three-system determined 
by points in the plane. All these quartics intersect in the critical curve (437). 
In like manner to a line in the p space corresponds the twisted sextic curve 
= ....... (475), 
the locus of united points of a two-system. 
The locus of Jacobian correspondents of points in the plane is the sextic curve 
[/(m). filM), = ^ .(476). 
Now any one of these sextics is the residual of the critical curve in the intersection 
of a pair of Jacobian quartics, and a curve meets its residual in t points, where 
r 4-^ = m(p + — 2).(477). 
In particular lor r — 40, m =10, p. v ~ we liave f = 20 ; and so there are 
2 Q 2 
