914 
Evidently it is of genus one (see § 3). Furthermore we have D=8 
(two.double pomts*on °°), he 12,48 = 0. 
The surface 2* (§ 7) containing the three curves *, w,’,w," has 
threefold points in 1 and 2. For, all orthogonal hyperbolas pass 
through these points. Two of these hyperbolas break up into the 
line /=1.2 and a chord 3,4 at right angles to it. 
EN 
points O consists of an w° containing the points Q, and an w'* con- 
taining the three other points ©; this follows immediately if we 
§ 9. If / has only point 1 with of in common the locus of the 
consider the points at infinity. 
We determine the order 14 of the latter curve independently by means 
of the number of times that one of the points O lies on /. The 
planes containing two chords at right angles in 1 envelop a cone 
of class siv; for the chord 12 is intersected at right angles by three 
chords and bears three planes in which the chords 18 and 14 are 
normal to each other (§ 8). So the triangle 1/4 is rectangular in 1 
for sie positions of .f and in each of these cases a point  coin- 
cides with 1. 
The chords of 9* intersecting / form a scroll of order five with / 
as double director line. So there are five chords normally cutting /, 
each case of which furnishes a point QO on /. So we find an w'* with 
sixfold point 1, through which point passes still a jiwefold secant. 
It is of genus one. as we can assign the point O, to the point & 
of o*. From m=14, D= 25, r= 28 (on / six tangents rest) we 
then derive B= 0, h= 52. 
The curve ow" has / as fivefold secant, is therefore rational and 
ot rank tens = 10,18 =O). 
Now the surface @2* has a threefold point in 1. 
§ 10. We still consider the scroll, locus of the lines of Eurer, 
ey, = My, Or, lying in the planes 4. 
Between the points of the curves u?° and w?° exists a correspond- 
ence (1,1). By projecting the corresponding points M and © out 
of an arbitrary line aq we generate a correspondence (20,20) between 
the planes of pencil (a). Of the 40 coincidencies 4 lie in each of 
the planes through « and one of the two points /, each of these points 
being fourfold point of 4? and of °°. In each of the other coin- 
cidencies lies a line e resting on a. So the scroll (e) is of order 32. 
We can verify this by means of the locus of the centres of gravity 
G of the triangles Jinn. It passes three times through each of the 
four points of o* at infinity and is therefore of order twelve. As 
