584 
passes through the line a,; this plane intersects the curve vy? in the 
point A’, in question. 
As the plane (4’,, a,) passes through the line «,, it is a tangent 
plane of the surface v’,, which contains the considered curve g?. 
Now the tangent planes of a ruled surface in the points of a gene- 
ratrix are projectively associated to the points of contact; the point 
of contact B, of the plane (A’,,@) is therefore harmonically separa- 
ted from the point A, by the points of contact of the planes a,, 
and a,,. As these two planes pass through the lines 6,,, and 6,,,, 
their points of contact B, and B,, are the intersections of these 
transversals with the line a,. 
If the surface ,,? describes the pencil (p,,’), the plane (4’, a), 
which touches the surface g’,, in the point B,, describes a pencil 
which is projectively associated to the pencil (g’,,). The figure 
produced by these projective pencils is a surface of the third order. 
To these planes of contact belong the planes a,, and a,, each of 
which is at the same time part of a degenerate surface @?,,; conse- 
quently these planes belong to the product and the rest is a plane. 
The figure produced by two projective pencils passes through the 
base-curves of these pencils. The plane just found contains therefore 
the line a, as this is the case with neither of the planes a@,, and «,,. 
It must also pass through the point B, as the intersection of a curve 
~’,, with its tangent plane in the point B consists besides of the 
line a,, of a generatrix through the point By. 
The locus of the points A’, which are associated to the point A, 
on the different curves g’* laid through the point A,, belongs to the 
intersection of the plane (4,,a,) with the surface y’,,, on which all 
these curves @° are situated and which passes through A,. This 
intersection consists besides of the line a, of a straight line 2; this 
is the locus in question. 
This line 2 passes through the point of intersection of the plane 
(B,,a,) with the line a,. Evidently this point of intersection is pro- 
jectively associated to the point B,, therefore also to the point A, 
The same must hold good for the intersection of the line 2 with 
the line a, If the point A, describes the line a,, the intersections of 
the line 2 with the lines a, and a, describe two projective sequences 
of points. Consequently the line 4 describes a guwadratic surface w’, 
the locus of all the points associated to the points of the line a,. 
To each of the lines a, and a, belongs a similar surface w’. 
§ 7. The points of the transversals },,, etc. are not singular points 
of the involution. For from the construction given in § 5 it follows 
