913 
tangent, i. e. of the line at infinity of A. So all the perpendiculars 
of the triangles /mn concur in J and / is a fourfold point of (0). 
But then the curve is of order twenty. 
If two points / coincide in a plane 4 the same happens with two 
points Ox. So through / pass eight tangential planes of @?° and as 
/ contains evidently szrteen points of this curve, w?° is of rank forty. 
It is of genus one on account of the (1,1)-correspondence between 
the point % of o* and Oy. 
From p=1, r=40, m= 20 and D= 24 (as there are two 
fourfold and four threefold points) we find (§ 5) B=0O, h= 146. 
So the curve has 146 apparent double points. 
§ 8. If l joins the points 1 and 2 of 9* the locus of the points O 
consists of three curves. For the points 0, and QO, always remain 
separated, if A rotates about / But on the contrary O* and 0' 
belong to the same curve; for the difference between the points 
3 and 4 disappears as soon as A is tangential plane. 
We now can determine the order of the curve (Q,) as follows. 
We look out in the first place for triangles 234 rectangular in 2. 
To that end we consider the-cubic curve 0’, which is the projec- 
tion of v* out of 2 on the plane at infinity. On each line through 
the trace 1, of 21 we determine the points //, separating harmo- 
nically the projections of 3 and 4 from the circle y?,. As e*, cuts 
the polar of I, in three points, 1, is threefold point of (//) and 
this curve a quintic. Its points of intersection with @*, are 1, count- 
ed thrice, six points on y°, and an other sextuple forming three 
pairs of traces of mutually rectangular lines 23, 24. So through 1,2 
pass three planes for which the angle 324 is a right one; therefore 
1 is a threefold point of curve (0). 
If line 34 is normal to 12, the point 0, lies on 1,2. So line 34 
generates a hyperboloid if 4 rotates round /; so by means of a section 
normal to 1,2 it is immediately clear that there are two chords 34 
at right angles to 12. 
So five points O, lie on /; therefore (O,) is a rational curve w,' 
of order sie with a threefold point. The line / is the bisecant of w," 
passing through the threefold point. Moreover we find r= 10, 
f= 7, = 0 . 
Evidently there are three positions of ./ for which 312 isa right 
angle; so the points 1 and 2 are threefold on the locus of the points 
O., 0, From this ensues that this locus is a curve w* with two 
threefold points. . 
As 34 happens to be tangent four times, w° is of rank sixteen, 
