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of o at infinity, the centres O,, O,, O, lie in the same direction at 
infinity and give rise to a threefold point at infinity. As we have 
found in the case of o' (§7) 7, contains two fourfold points of 
(O). But I, bears two points Q more, originating from the two 
trisecants of of meeting /. For if in a plane / the points 1, 2, 3 
are collinear, the three perpendiculars of the flattened triangle 124 
are parallel. Then the four orthocentra lie on the normal ¢ through 
4 on the trisecant; so q is quadrisecant of the curve (Q) and the 
order of (O).is 22. . 
There are six tangents of of meeting / and therefore as many 
tangents of w** doing likewise; as w°* has evidently 18 points in 
common with /, this curve is of rank 42. As it corresponds in genus 
to o* and its singular points are equivalent to 24 double points, we 
find by means of the formulas given above B= 0, h = 186. 
§ 18. If / contains the points 1, 2 of 6*, the ioeus (V) breaks up 
into three different curves. As in $ 8 we find here through 1, 2 
three planes bearing chords 23, 24 normal to each other, so 2 is 
threefold point of (0). 
Jut now the line 34 describes a cubic scroll (with double line /) 
if A rotates about /; so 12 is cut orthogonally by tree chords. 
So we find for (O,) and (0) two rational curves of order seven. 
The locus of O, and 0, is once more an w* with two threefold points. 
The three curves are situated on a surface 2’ forming the locus 
of the orthogonal hyperbolas 1234. For. in the three planes 4 bear- 
ing a chord 34 normal to /= 12, the hyperbola degenerates into 
these two chords and 7; so / is threefold line from which ensues 
moreover that 1 and 2 are fourfold points. 
So we may conclude that for an arbitrary position of/ the corre- 
sponding orthogonal hyperbolas form a surface of order five with / 
as threefold ‘line. 
Let us still consider the case that / is a frisecant, containing the 
points 1, 2, 3 of o*. Then U, is always at infinity and each of the 
remaining three points O describes its own curve. 
If 4 coincides with 1, 0, is at infinity, which also happens if A 
contains a point of of at infinity and if A touches y*,,. From this 
we conclude that each of the points 0,, O,, O, describes a rational 
curve of order seven, with threefold points in two of the points 1, 2, 3. 
In fact each of the points 1, 2, 8 is vertex of a rectangular tri- 
angle for three positions of 4, or more exactly of two suchlike tris 
angles; for, if 14 is normal to the trisecant, 1 is orthocentre of 124 
and of 134, 
