(579) 
so it is of order 2. Its points of contact with (¢ x) are determined 
on (oa) by the nm rays s corresponding to the 7 rays ¢ through the 
point (or 2). 
If x passes through one of the 2(n— 1) points of intersection 
of 5 with the envelope tr, two of the points of contact of (o ze) coincide. 
Regarded as locus of points («) then consists of a curve of order 
(2n —1) and the right line (62). 
The planes containing curves of the complex for which two points 
of contact of the multiple tangent coincide form 2(n—1) sheaves 
having their vertices on the line of intersection of 6 and t. 
If « passes through a ray s,, then (2) as envelope consists of a 
pencil having its vertex in the trace of the homologous ray ¢, and 
of the pencil (S, x) of which each ray belongs 7 times to the complex, 
because it is intersected by 7 rays ¢. As locus of points (a) is here 
the line connecting the vertices of the pencils counted 27 times. 
If z contains a ray ¢, the envelope (2) consists of a pencil having 
the trace S, of the homologous ray s, as vertex and of a curve of 
class 7 for which (oa) is an (n—1)-fold tangent. As figure of order 
2n the curve (x) breaks up into a curve of order 2(m—1), its (n —1) 
fold tangent and the tangent which can moreover be drawn to it 
out of S,. 
If one brings a through one of the coincidences (;, then (a) breaks 
up in the same way into a pencil with vertex C, and a curve of 
class 7. 
The complex possesses an n-fold principal point S and (n +1) 
single principal points Cy. 
§ 4. Let us now consider the surface of the complex A of an 
arbitrary right line /, thus the envelope of the rays of the complex 
resting on /. The rays in a plane a brought through / envelop a 
curve (a) of order 2n ($ 3). If a is one of the 2» tangent planes 
through / to the cone of the complex of the point P lying on J, 
then two of the tangents drawn out of P to (a) coincide, so that 
P is a point of (xr). So each point of / belongs to 2” curves of the 
complex; consequently / is a 2n-fold right line of 4. 
The surface of the complex is of order 4n. 
In the planes connecting / with the principal points C‚ the curve 
(a) breaks up into a curve of order 2 (7 —1) and two right lines. 
This also takes place when z passes through one of the rays t 
resting on /. In the plane through / and S the curve (a) degenerates 
into a right line to be counted 27 times. 
In each of the planes connecting / with the points of intersection 
