509 
of £? intersects the demanded surface in each of its two supporting 
points with #° in eight points and no more; because two chords of 
k* cannot intersect each other outside 4£°; the demanded surface is thus 
of order sixteen, and it has k* as an eightfold curve, k* as a single curve. 
That 4° is the only manifold curve follows again out of the circum- 
stance that two chords of 4° can meet each other only on the curve 
itself; on the other hand the twenty common chords of k* and k* are 
again double generatrices. As an eightfold point counts for 4.8.7 — 28 
double points, the complete number of double points of a plane section 
is 3.28 + 20—104; a plane curve of order sixteen can however 
contain at most 4.15.14 105 double points; so the surface is of 
genus 1. 
10. Through a point P of space pass two chords > of 4‘ situated 
in the plane a through P and the line of intersection s of the two 
polar planes of P with respect to the two quadratic surfaces ®, , d, 
(§ 2) intersecting each other in /4*; we shall conjugate this plane a as 
focal plane to P and we shall discuss the focal system that is formed 
in this way. Lach point of space has then one focal plane so 
a=1")), with the exception of the points of k* having ow focal 
planes, viz. all the planes containing the tangent in that point. 
In order to find inversely the number 3 of the foci P of an arbi- 
trary plane a, we intersect that plane with ®, and @®,; this gives 
rise to two conics k,’?, 4,7, and with respect to these.we take the 
polar lines p,,p, of an arbitrary point P of z. The polar planes of ? 
with respect to ®,, ®, then pass through p,,p, and the line s con- 
jugated to P contains the point of intersection of p, and p,; if s is 
thus to be situated in plane a, then p, and p, must coincide, and this 
takes place only for the vertices of the polar triangle which /:,* and 
k.* have incommon: so 8 is == 3. 
The third characteristic quantity, y *), indicating how often a focus 
P lies on a given line, whilst at the same time the focal plane zz 
passes through that line, is found as follows. When P describes 
the line / the two polar planes rotate round the two lines /,, /, con- 
jugated to / with respect to ,,,; their line of intersection s 
describes a regulus with /, ,/, as bearers, and passing through the 
vertices of the four doubly projecting cones of £*; this regulus intersects 
lin 2 points, through which every time one line s passes, and the 
foci conjugated to these lines lie on / as is in fact the case for all 
lines s of the regnlus; for these two foci however the focal plane 
a = Ps passes through /; so y = 2. 
1) Sturm, “Liniengeometrie” I, p. 78. 
