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therefore with Q and we see that the curve o passes through @ 
and here touches at the line ¢. 
A plane JV passing through the point Q, intersects the above 
mentioned cubic cone along three generairices, which each contain 
one point S. The point QQ and these three points S are the inter- 
sections of the plane V with the curve o; this curve is consequently 
of order four. 
§ 7. It appeared in § 3 that, if 7’ is a node of a surface a’, this 
point must be a singular point of S*. for if / is an arbitrary straight 
line passing through 7’, the said surface a* will have two points in 
common with the straight line 7 in 7. If we take for the line / 
one of the tangents of the surface a° in the point 7, two of the 
surfaces touching at / will coincide with the said surface a? and 7 
is consequently a coincidence. The two other points of the associated 
group are the intersections of the-line / with the surface 77°, which 
belongs to the point 7. These tangents / form a quadratic cone, 
which intersects the surface MF? along a curve of order ten. To this 
curve, however, belong as may be easily seen the 6 straight lines 
passing through 7 and lying on the surface a*. The rest-section, i.e. 
the locus of the points of the above mentioned groups, is therefore 
a curve of order four. 
§ 8. The points that belong with an arbitrary point P to the 
same group S*, form a curve c° of order five. If now the point P 
describes a line /; these curves will describe a surface / of which 
we shall determine the order. 
For this purpose we investigate the intersections of 4 with the 
surface 11°, belonging to as point P of the line /. 
These surfaces HZ’ form a pencil. For, through an arbitrary point 
X passes one surface a’, and the tangent plane in X at this surface 
intersects the line 7 in one point P, which with X belongs to a 
same group of S*. Through this point only one surface JT passes. 
The last reasoning does not hold good if X is chosen on the base- 
curve 9°; it then lies viz. on oo’ tangent planes of surfaces a°. The 
curve 9° is therefore a part of the base-curve of the pencil (17). 
Neither does that reasoning hold good if the said tangent plane 
passes through the line /. The rest of the base-curve of the pencil 
(HI) is therefore the locus of the points of contact of the tangent 
planes at surfaces a’ passing through the line /. This curve must 
be of order 16, as it forms, together with 0°, the base-curve of the 
s 
pencil (41°. This is really so, for a plane V passing through / inter- 
