1150 
be laid moreover, passing through the tangential point 7 of B,; 
the points of contact C of these planes we call the cotangential points 
of B,; they form a surface y. 
The surfaces t° belonging to the points B, and 5, intersect along 
a curve of order 49, whieh has 18-fold points in the points 6, and 
B. 9-fold points in the other base-points. A * of the net inter- 
sects this of” outside the base-points, moreover in 8 points, and 
consequently contains 8 curves e*, on which B, and 5, are cotan- 
gential. The tangents of these curves in B, form therefore a cone 
of order eight, so that B, is an octuple point of the surface y. The 
same holds good for the points b,..., B, 
As none of the cotangential points can coincide with 5,, y does 
not pass through B,. 
From this we may easily deduce that y is a surface of order sixteen. 
§ 6. Out of an arbitrary point P twelve osculating planes may 
be laid to each of the curves 9*; the locus of the points of contact 
of these planes we call the polar surface II of P in regard to the 
congruence |9*|. 
This surface will again have multiple points in the base-points Be. 
The tangents in the point ZB, are the tangents of the curves 0‘; 
the osculating plane of the point B, passes through P, consequently 
through the line P5,. Now we have found in $ 3 that there are 
two curves o*, which osculate in B, an arbitrary plane passing 
through that point; there is further one 0*, which touches the line 
PB, in B. An arbitrary plane passing through PB, contains 
therefore three of the tangents wanted, so they form a cone of order 
three. The points Bj are consequently triple points of the surface 11. 
From this it follows that MZ is of order nine. 
$ 7. Each of the osculating planes considered in the preceding 
$ intersects the o* to which it belongs moreover in one point; the 
locus X of these points we call the satellite of the point P. 
The tangents of this surface in the point B, are the tangents of 
those curves @*, in which one of the 9 oseulating planes laid through 
B, passes through P, consequently through the line PB,. The order 
of the cone formed by these tangents is equal to the number of 
these v*, which are lying on an arbitrary #* of the net. 
If we now project all the v* lying on one and the same @?, out 
of B, on a plane V passing through P, we acquire as projection 
a pencil of plane cubics. The base-points of this pencil are the pro- 
jections of the points 5,....8, besides the intersections of V with 
