485 
two rays f lie in 8,,,; the remaining five are transversals to a,, 
and b 
Hence the rays of the bilinear congruence whieh has a,, and 6,, 
for directrices, can be arranged into oo? groups of five lines {/. 
5. If the ray ¢ describes a pencil (Zr) the nine corresponding 
rays ¢’/ remain on a ruled surface (¢’); the degree of this surface 
we can determine by investigating the number of lines ?’ resting 
on A, A, 
Consider therefore, to begin with, the rays ¢’ meeting a,, outside 
A, and A,. Such a line # can be chord to an «° composed of a,, 
and a conic a? in the plane a,,,. Of the pencil (4) the particular ray 
which rests on a,, meets on the intersection of the planes @,,, and 
ron @ and is therefore a chord to a composite «*. Together with 
the 8’ which has ¢ among its chords this «* determines six lines 7’ 
resting on a,, and, in addition to this, three lines ¢’ in the plane 
Oras 
Similarly the plane «,,, contains three lines ¢t’; these are common 
chords to an «° and a 8°, the first composed of a,, and a conic in 
@,,,, the latter having among its chords the ray of the pencil (Zr) 
which meets a, 
In the same way the planes «,,, and a,,, too contain tree rays 
v’ each of the ruled surface conjugated to (7). 
6. In order to determine the number of rays ¢’ passing through 
A, we consider the surface A engendered by the curves «° having 
each one ray of the pencil (7, rt) among their chords. 
Let d denote the line of transit of @,,, in the plane r, D the 
point of transit of a,,. The ray of (7,1) which meets a,, determines 
on & a conic, which lies in @,,, and in combination with a,, con- 
stitutes an «° belonging to the above-mentioned surface A. It follows 
from this that 4 passes through the ten lines aj, and contains ten 
conics, one in each of the planes Bor Hence the intersection of A 
a,, and a conic passing through 
and «@,,, consists of the lines a, a, 5 
12 
A,, A,, A,, so that 4 is a surface of degree five. *) 
An arbitrary plane ® is intersected by the curves a? in the 
triplets X,, X,, XN, of an involution (XY). Since through any two 
curves « a quadrie surface can be laid, it is possible to join any 
two triplets of the aforesaid involution by a conic. Hence a point 
') 4° evidently has a triple point at each of the five points Ag. The locus of 
the pairs of points at which each 4? of /,° meets the corresponding ray ¢ is a 
curve c* having a double point at 7’ Hence 4° is intersected by r along an 
additional line /. It follows from this that /° is also the locus of the curves «3 
which meet 1. 
