944 
secting a, and /. Hach of the other twelve pairs consists of a trans- 
versal of 7, ar, a and a transversal of J, am, dy perpendicular to it. 
3. Through any point P pass six curves of the congruence [07]. 
For the locus of the 0’ which have CP as a chord and which rest 
on the lines a, has CP as a sixfold straight line. 
Any point 4, of the line ag is singular. The curves 9° through 
A; form a monoid O* with vertex Ay; and fourfold straight line 
AC. It. contains fourteen pairs of lines arising in the following 
way. Three pairs consist each of a transversal through Aj to a, am 
and a straight line intersecting a, and /== AC. Two pairs consist 
each of a transversal of J, a, a, a, and the perpendicular out of 
Ax to this transversal. In order to find the other pairs we consider 
the cone formed by the perpendiculars 6; out of Az to the trans- 
versals of /, a, dn. As two of these transversals are perpendicular 
to J, bp coincides twice with /. The cone in question is therefore 
cubical and has / as a double generatrix. Consequently there are 
three orthogonal pairs of lines of which the line 6; passes through 
Ar. In this way the nine remaining pairs are found. 
4. Also the point C is singular. The determination of the order 
of the surface 7° formed by the curves 0? passing through C, comes 
to the determination of the number of orthogonal hyperbolas through 
C resting on five straight lines 1, 2, 8, 4, 5. Using the principle of 
the conservation of the number we can suppose the straight lines 
1, 2,and 3 to lie in a plane y. Through C and the point 12 pass 
four o*, resting on 3, 4, and 5; analogously we find four of them 
through C and 23 and four through C and 13. 
All the other figures satisfying the conditions are pairs of lines 
of which one line, s, lies in p, while the other, ¢, passes through 
C. To these belongs in the first place the line s in p intersecting 
4 and 5, in combination with the perpendicular ¢ out of C to s. 
Let us now consider the plane pencil (s) in p which has the 
intersection M of 4 for vertex. The perpendiculars out of C to the 
rays of (s) form a quadratic cone; the two generatrices ¢ resting 
on 5, belong each to an orthogonal pair of lines (s,f). As we can 
interchange 4 and 5, the group considered contains four pairs (s, 2). 
Finally we find the figure formed by the transversal ¢ through C 
to 4 and 5, combined with the line s in p cutting it at right angles. 
In all we found 3x 41142 X<2-+1=18 figures 07; the curves 
o? through C form consequently a surface ['°. 
