946 
class, for through D, passes also the line r at D, perpendienlan to 
tbe ray ¢ of which D, is the intersection. 
Finally we have to consider the case that the line 7 rests on 4, 
5 and 6. If we now also project the scroll (4)? orthogonally on 
and draw through the intersection 7’ of tand p the line 7 perpen- 
dicular to ¢, 7 envelops, as appeared above, a curve of the fourth 
class. From this follows, that also the plane (7/) envelops a curve 
of the fourth class, so that through C there pass four planes in each 
of which a transversal of 4, 5, 6 is cut perpendicularly by a trans- 
versal of 1, 2, 3. 
In all we found 3x6+3-+3«3+4=— 034 figures 0°; the 
locus of the o? resting on a straight line /, is consequently a surface A**. 
The curve 0? in the plane (CU) is apparently a double curve. The 
four lines a are sixfold on A; for the curves 0% through a point of 
a form a surface O°. 
7. The planes Ca; may be called singular because they contain oo' 
orthogonal hyperbolas. This will also be the case when a plane 
through C cuts the lines ap in an orthocentrical group. Now the 
orthoeentres of the triangles A, A, A, of which the planes pass 
through C, form a surface; there must therefore be a finite number 
of singular planes of the kind in question. 
In order to determine this number, we first consider the locus of 
the orthocentre H of a triangle CA,A,, when A, lies on a,, A, on 
a, The plane through a point A, perpendicular to the ray 4,C 
contains one point A,, hence one triangle 4,4,C' of which A lies 
in A,. Consequently the surface in question contains the straight 
lines a, and a, 
In the plane Ca, lie oo! triangles A,A,C; their orthocentres lie 
in a conic H? through C and the intersection D, of a,. The intersection 
of the surface with Ca, consists of a, and H?; we have therefore 
a surface H*. Three times H lies on a,, or, in other words, through 
C pass three planes in which the orthocentre of A,A,A, lies in C. 
We consider now the surface formed by the orthocentres of the 
triangles A,A,A, of which the planes pass through C. 
If H is to get on a,, A,A, must be perpendicular to A,A,. In 
each plane through a fixed straight line A,C we draw through A, 
the line / perpendicular to A,A,. If this plane is perpendicular to 
A,C, | coincides with A,C; hence / describes a quadratic cone. Two 
of the generatrices mieroet a,; through A,C pass consequently two 
planes in which AH goede with A,. But then a, is a double 
straight line of the surface in question. 
