503 
of rays in a plane; in order to find this we have but to determine 
the number of inflexions of a plane section of 2°. We have already 
seen that this plane section is of order 6 and of class 24, and that it 
contains 3 double points, whilst the number of cusps is 0; from this 
ensues easily that the number of inflexions is 54, the number of 
double tangents 192; the congruence of the principal tangents of 
2" has therefore the characteristics 84 and 54, those of the double 
tangents 5202 and 192. 
7. Through each point P of 2° passes a plane a, in which are 
situated one chord a of #* and two chords 6 of £*; we wish to 
study the surface which is enveloped by those planes a. The class 
of this surface can be determined in different ways; we shall deduce 
this number in the first place by asking how many planes a pass 
through a chord a of £°. Through the point of intersection A, of 
a with &* passes one plane a which in general however does 
not pass through a, and the same holds for the second point of inter- 
section <A,. Besides these two points a has still but 2 points S,, S, 
in common with 2°, and through these passes a plane 2 containing 
a; for S, eg. is a point of @° exactly for this reason that the 
chord a lies with two chords 5 of kf in a plane a. So to each 
of the two points S,,,S, a plane a through a corresponds. 
However planes a can also pass through a without it being 
necessary for the point of intersection P of the triplet a + 25 to 
lie exactly on a itself. If we make a plane « to rotate round a, it 
contains in each position 2 more chords a and 6 chords 6, forming 
a complete quadrangle. The two chords a describe the two quadratic 
cones by which #° is projected out of the two points A, ,A,, the 
diagonal points of the complete quadrangle describe a twisted curve 
possessing in each plane « three points apart from the points lying 
on a itself and which are nothing but S, ,.S,; so the diagonal points 
form a twisted curve 4° of order 5 resting in 2 points S,,S, ona, 
(and containing evidently the four vertices 7,,.., 7,, § 3). Let us 
consider a point of intersection of this 4° with one of the just men- 
tioned quadratic cones, we then have evidently obtained a point of 
2° and at the same time a plane a through a. Now &° intersects 
each cone in ten points, but among these are S, and S, ; so outside 
a lie only sixteen points of intersection and if we again add S, and 
S,, counted once, we then find that the surface enveloped by the planes 
x bearing a triplet a+ 2b is of class eighteen. We shall indicate 
Troy 82,5 | 
As easily we can determine the class of 2,, by means of a chord 
