1149 
through B, consequently in the case considered on / and on a 
second straight line /’. If one of these intersections is made to 
approach to B, along the associated generatrix, one curve g° is 
obtained every time, which oscu/ates the plane V in B,. The gene- 
ratrix mentioned is the tangent in this point, while the tangential 
point of B, lies on the second generatrix. In order to find the 4 
wanted, which passes through the point 7’ we have to make the 
intersection with / to approach to B. In this way one definite 
curve ef is found, consequently one point 7’. 
The straight line / intersects therefore the surface rt outside 5, 
only in one point, this surface is therefore of order seven. 
A curve ef intersects t’, besides in the point 4, and the associated 
tangential point, only in the other base-points #,..... B,. From this 
it is easy to see, that these points are triple points of t’. 
From this it ensues at once: If we consider all the curves 0‘, 
on which B, is the tangential point of B,, the tangents of these 
curves form in each of the points 4,.... B, a cone of order three. 
A surface ® contains three of these curves; for the plane of 
contact of dP° in the point B, has three generatrices in common 
with the cone mentioned, and each @*, which touches d>? in one 
of the base-points lies entirely on this surface. From this it ensues 
at once that in consequence of this the tangents in the point 5 
form a cone of order three, for of these tangents three lie in the 
plane that touches ®? in the point 5, 
$ 4. Through the base-point 5, nine other osculating planes may 
moreover be laid to each of the curves o*. The points of contact 
A of these planes we shall call the antitangential points of B, ; 
they form a surface a. 
If B, is a point of inflexion, one of these planes of osculation 
coincides with the one in B, so one of these points with B,. As 
above it follows that B, is a sextuple point of a. 
The surface « will also pass through the point B,; the tangents 
of « in this point are the tangents of the curves e*, on which 5, 
is the tangential point of B, It appeared above that they form a 
cone of order three; B, is consequently a triple point of a. The 
same holds good for the other base-points. 
A curve o* intersects the surface a, besides in the point B 
moreover in the 9 points A lying on 9%, a is therefore a surface 
of order nine. 
5. To each of the curves of eight other osculating planes ma 
Q eg Sl y 
