945 
5. Any ray through C is a chord of sir 07, belonging to IT; 
hence Cis a twelvefold point. 
The transversal 6,, through C to a, and a, is cut at right angles 
by two transversals of a, and a,; the sev lines bj, are accordingly 
double lines of T. To them 12 single lines are connected. 
To each ¢,,, of a,, a, a, we draw the perpendicular 5 out of C 
and we consider the cone which has the straight lines 6 as genera- 
trices. Let y be a plane through C and a straight line c of the scroll 
to which a,, a,, a, belong. Through the intersection D of t,,, we draw 
in y the straight line d perpendicular to c. As c is cut at right 
angles by two lines ¢,,,, d cvincides twice with c, envelops conse- 
quently a curve of the third class with double tangent c. The three 
lines J meeting in C are generatrices of the cone (6); this is con- 
sequently cubical and there are three pairs of lines (6, ¢,,,). In all 
we find twelve pairs of lines o* of which one of the lines rests on 
three straight lines a. 
Finally there lie on I the two transversals 6 
to a straight line through C. 
Each of the four o? which have a line a as a chord, is a double 
curve of I. 
i234 each connected 
6. To find the order of the surface A formed by the 0? resting 
on a straight line /, we try to find the number of curves 0%, in 
planes through C, which rest on six straight lines 1, 2, 8, 4, 5, 6, 
and again sappose 1, 2, 3 to lie in a plane gp. 
Through the point 12 pass siv 0? resting on 3, 4, 5, 6, while their 
planes pass through C. Analogously sir pass through 28 and sir 
through 13. All the other figures degenerate into a straight line s 
of p and a line ¢ cutting it at right angles. 
The plane through C and the intersections of 4 and 5 with » con- 
tains a figure (s,¢) of which the line ¢ rests on 6. We obtain here 
a group of three pairs (s, 2). 
If s is to pass through the point D=(4,), ¢ must rest on 5, 6 
and (CD, The orthogonal projections ¢’ of the straight lines of the 
scroll (/)* envelop a conic, Let the perpendicular 7 out of D, to ¢t/ 
be associated to the ray s joining D, with the intersection 7’ of a line 
t; r being perpendicular to two lines ¢’, hence associated to two 
rays 8, there are three coincidences += s. We find therefore three 
pairs of lines (s, 4) satisfying the given conditions; in all a group of 
3 X 3 figures 07. : 
From this it ensues at the same time, that the straight line r 
cutting the ray fin 7’ at right angles, envelops a curve of the fourth 
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