1151 
the two generatrices of @° passing through B,. Any osculating plane 
of a curve o° laid through B, intersects the plane V along a tangent 
in a point of inflexion of the projection. Now the inflexional tan- 
gents of a cubic pencil envelop a curve of the ninth class; conse- 
quently nine of the planes of osculation wanted pass through P. 
So it is proved that Y has the base-points B as nonuple points. 
From this it easily ensues that + is of order twenty-one. 
$ 8. Through an arbitrary point P passes one curve 9; let P’ 
be the tangential point of F lying on this 9%. As to each point P’ 
of a ef nine antitangential points P belong, there exists a corre- 
spondence (1,9) between the points P and P’. 
If P is a point of injleaion of the o* the points P and P’ coin- 
cide; the surface of coincidence of this correspondence is therefore 
the surface r'® found in $ 2. 
If P describes a straight line /, P’ will describe a curve gy. The 
latter has nonuple points in the base-points Bj; for every time P 
comes on one of the surfaces «° found in § 4, P’ will lie in one 
of the base-points. 
A surface ®° intersects gy, further in the #vo points, which cor- 
respond with the two intersections of ®* and /. So we find that 
gi is of order 37. 
If P describes a plane V, P’ describes a surface Py. This is 
intersected by a 4° along the locus of the points P’, associated to 
the points of the intersection of ®* and V. As this locus is of order 
74, Py is of order 37. 
A g* intersects ®y*’, besides in the points Bj, in the 4 points //, 
associated to the 4 intersections of o* and VV; #7”, has therefore 
eighteen-fold points in the points Br. 
If P’ deseribes a straight line /, P will describe a curve wy, 
which has septuple points in the base-points Bj; they correspond to 
the intersections of / with the surfaces t° found in § 3. A @? inter- 
sects y, further in 18 points, corresponding to the two intersections 
of ® and /; w too is therefore of order 37. 
If P’ describes a plane V, P describes a surface wy. Just as 
this was done above for dy, we find that wy is of order 37 and 
has fourteen-fold points in the points Bg. 
§ 9. If to each point P are associated the eight points which lie 
on the o° passing through P and are cotangential with P, we get 
an involution I? in space. 
