912 
$ 5. In the case of an arbitrary line / the locus of the points M 
is a twisted curve °°. Any plane A through a point at infinity of 9° 
furnishes three points of F‚ lying in different directions, and on 
y?,, the curve has two fourfold points. Moreover it possesses three 
fourfolds points in the cyclic planes through l. 
As any plane 4 bears four points M/ none of which generally 
lies on /, u?" has with / sixteen points in common. Each of the eight 
tangential planes of o* furnishes a tangential plane of uw; so this 
curve is of rank forty. 
It is of yenus one, for one can assign the point My, to each point 
k of o*. So the generally known formulas 
r= m(nm—1) — 2 (h4+-D) — 38, 
p = 4 (ml) (m—2) -- (44 D8), 
where we have r=—40, m=20, D=.30, p=1, give @=0O, 
h = 140. 
So the curve has no cusps, but 140 apparent double points (bise- 
cants through any point). 
$ 6. If the points 1, 2, 3, 4 of gf form an orthocentric group, 
their plane A cuts all the ®* according to orthogonal hyperbolas ; 
then all the planes parallel to 4 furnish orthocentric groups. 
The planes cutting a director cone of #* in two edges normal to 
each other envelop a cone of the second class. So two concentric 
director cones determine four planes cutting the -two corresponding 
® and therefore all the #* of the pencil in orthogonal hyperbolas. 
From this ensues: there are four systems of parallel planes cutting 
e* in orthocentric groups. 
§ 7. We consider in any plane 4 through / the orthocentres Of 
of the triangles /mn, which four points lie with the points 1, 2, 3,4 
on an orthogonal hyperbola w?. 
Evidently w? is the section of 4 with a @ through of; now we 
can bring through / a second plane cutting that ®* in an orthogo- 
nal hyperbola ($ 6). So any point of / lies on two curves o’, ine: 
/ is double line of the locus of the curves w?. Therefore: the locus 
(O) of the orthocentra Oj lies on a surface 2' with double point 1. 
In order to determine the degree of (V) we remark that in a 
plane 4 through a point at infinity of g* three points O lie at infi- 
nity in the same direction, which proves that TP, contains four 
threefold points of (U). If A touches the circle y’,, the point of 
contact is separated harmonically by 7°, from any point of the 
