800 
a pencil of planes having the directrix through O as axis; all the 
curves of intersection of , with the planes of this pencil pass 
therefore through the same point S’ of @,. These intersections 
correspond in S, to conies through A,, A,,A, and the point S cor- 
responding to S', which point S lies on c, because S' is a point of q. 
The points B,, B,, B, are accordingly the intersections of the 
straight lines PA,, PA,, PA, with a conic of the pencil through 
S, A,, A,, A, 
In order to determine the point S, we remark that the directrices 
of R, cut the generatrices p and q in projective point ranges; three 
pairs of corresponding points are the intersections of p and q with 
Py Po Ps The directrix through QO and with it the point S’ are 
therefore found by determining the point of g corresponding to O. 
In the image of ®, the directions round P are therefore projec- 
tively conjugated to the points of c, and that in such a way that 
both the points of intersection B',, 5',, B', of these straight lines with 
, correspond to the directions PA,, PA,, PA, If we project the 
latter points out of A,, there appear round P and A, two perspec- 
tive pencils of rays of which the axis of perspectivity is found as 
the join of B’, and B’, If this cuts A, A, in S" the second point of 
C 
intersection of A,S" and c, is the required point S. 
Any point P of A,A, defines out of the fourfold infinite linear 
system S, through A,,..., A, an S, in which one point S has been 
constructed; to each point P of A, A, belongs therefore one point 
S, or one point S". 
Let us now try to find the number of points P belonging to one 
point S or S". When P varies, B’, and B’, describe an involution 
on c,; the envelope of B’,B’, is a conic k, touching c, in the points 
A, and A,*). Out of S" we can draw two tangents to this conic, 
which define two pairs of points on c,, hence two points P, and 
P, on A,A,. The relation between P and S is therefore a (2,1) 
correspondence. 
Now it is known from § 2 that the curves of the pencil contain- 
ing the degenerations PA, PA, PA, have all a point of inflexion 
in P and also a common harmonical polar line for the pole P. 
The harmonical polar lines of all such pencils out of S, must pass 
through the 4 harmonical point P’ to P with respect to A, and 
A,; also the polar straight lines of P relative to each of the conics 
of the pencil (S,A,,A,, A,) must pass through P’; P and P’ are 
ij R. Sturm, "Die Lehre von den geometrischen Verwandschaften” 3ter Band, 
S. 138. 
