1155 
$ 3 they are 8 eurves of order 14, which have a sextuple point in 
the associated intersection and triple points in each of the others. 
So we see that af is a 27-fold curve of r,. 
The intersection of +. with a quadric is therefore of order 
148 44 Xx 27=8 x 144 4 X 36 = 256; r, is therefore of order 128. 
The antitangential points of P describe a surface A,. It is inter- 
sected by a surface a®, besides along «*‚ in a curve of order 
437 —=148, having 28-fold points in the intersections of a* and 
B's B' is therefore a 28-fold curve of A,. A surface 6? intersects A, 
besides along 8%, in 8 curves of order 18, which each have one of 
the intersections of 5? and «@ as sextuple point and the remaining 
ones as triple points; «* is therefore a 27-fold curve of A,. The order 
of A, therefore amounts to: 74+2 xX 27=8 X9+42 x 28= 128. 
The cotangential points of P describe a surface L. It is inter- 
sected by a surface «°, besides along «°‚ in a curve of order 
4 72 = 288, which has 64-fold points in the intersections of a* 
and pt; 8* is therefore a 64-fold curve of I. A surface 6° inter- 
sects I, besides along 8°, in 8 curves of order 32, each having 
8-fold points in 7 of the 8 intersections of 4° and a‘; «“ is therefore 
a 56-fold curve of EF. The order of TM, amounts consequently to 
1444-2 56=8 X16-+4+ 2 x 64= 256. 
§ 12. Ifa point P describes an arbitrary plane J’, the tangential 
point P” of P describes a surface ®y. This is intersected by a surface 
a’, besides along «°, in a curve of order 2 < 387 = 74, which has 
18-fold points in 8 intersections of a* and 3*. Consequently «* and 
8: are 18-fold curves of ®y, and the order of ®j amounts to 
ea en laf 
The antitangential points of P describe a surface Wy. This is 
intersected by a surface a’, besides along «', in a curve of order 
2 37 = 74, which has 14-fold points in the intersections of a? and 
8. Consequently ae“ and 8* are 14-fold curves of Wy, and the order 
of Wy amounts to 37 +2 14=— 65. 
The cotangential points of P describe a surface Zy. This is inter- 
sected by a surface a’, besides along a‘, in a curve of order 
2 72 —= 144, which has 32-fold points in the intersections of a? and p*. 
Consequently «* and g' are 32-fold curves of Ey, and the order 
of Ey amounts to 72+2*32=136. 
