1136 
In order to find the locus of the associated points P we observe 
that, if a line / describes a plane pencil, the points of the group 
of S' lying on / will describe a curve of order five. The inflectional 
tangents PQ envelop a curve of class 9; the points of the group 
lying on PQ consequently deseribe a curve of order 9 X 5 = 45. 
To them belongs the curve «°°, twice counted, as in Q two points 
of a group coincide. The rest curve, ie. the locus of the points 
P, is therefore of order 21. 
This curve is the intersection of the plane | with the surface ®. 
This surface is consequently of order 21. 
The curve «* is the intersection of the plane V with the surface 
y. Now, however, an arbitrary point Q of the surface w belongs to 
one point P of the plane V, while the point Q of the curve v? 
belongs to two points P. The curve «°° is therefore a nodal curve 
of the surface yw. This surface is therefore of order 24. 
The order 21 of the surface gm gives the number of times that 
the point Q lies in a plane V and the point P on an arbitrary 
line /. It consequently also gives the order of the curve described 
by the point Q if the point P describes a straight line /. 
In the same way, if the point Q describes a line /, the point ? 
will describe a curve of order 24. : 
§ 6. If a line / passes through a point Q of the base-curve 0°, 
two of the surfaces a*, which touch at /, will coincide into the 
surface a*, which touches / in Q. Any secant of 9’ therefore also 
bears a coincidence of S*. 
Such a secant is touched outside g* by two surfaces a’; the points 
of contact are associated to Q by S*. If one of these points of contact 
coincides with Q, three associated points of the S* will coincide in 
(J. The surface a* belonging to this point of contact has in Q 3 
coinciding points in common with / in that case. The principal 
tangents passing through a point Q of the curve 9’, form a cone 
of order three; for, a plane JV passing through the point Q, inter- 
sects the pencil (a*) along a pencil that has a base-point in @, and 
the curve «'*, which is the locus of the points of inflection of the 
curves of this pencil has a triple point in Q. 
On each generatrix of this cone lies another point S, which is 
associated to Q by S*; these points form a curve 6, passing once 
through Q. For let us consider the tangent ¢ in Q at the curve 9’. 
An arbitrary surface a* intersects the tangent t apart from Q only 
in one point, so™there is not a single surface a’ that touches at ¢ 
outside Q. The four associated points of S* lying on ¢ coincide 
