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12. In the null-system (Cc) P has a null curve of order (5n—8) 
with node P (§ 7). Of its intersections with the null curve with 
respect to the system (/’,/), 10 lie in P. They also have the uni- 
cuspidal points U in common, for which the tangent w passes through 
P. In each of the remaining (5n»—8)*—10—(10n?— 25n+-12) inter- 
sections G, a cuspidal curve has with its tangent g four points in 
common. From this it ensues that the four-point cuspidal tangents 
envelop a curve of class (15n*—55n-+42). 
If n is equal to three, the curves y* with four-point tangents are 
replaced by conics, each with one of its tangents. The null-system 
(ff) then has the characteristic numbers 5 and 2; the null-curve 
(P)’ of P is of class 22, consequently sends 12 tangents f through 
P, and each of these straight lines forms with the conic touching it 
a y° with four-point tangent. In conformity with this, the form 
15n*—55n-+-42 produces for n = 3 the number 12. 
13. In a quintuple infinite system S©) each point D is node for 
a net of nodal curves. A straight line d passing through D deter- 
mines in it a pencil, of which all gr touch at d in D. There is 
consequently one cuspidal y", which has a straight line c passing 
through D as cuspidal tangent. The curves y”, with cusp D, form 
a system with index two, for the curves J”, passing through any 
point P, form a pencil, which contains two curves with cusp in D. 
If every straight line c passing through D is made to intersect with 
the cuspidal 7”, which it touches in D, there evidently arises a 
curve of order (n-+-2), which has a quintuple point in D. From this 
it ensues that five cuspidal curves have in D a cusp, where the 
euspidal tangent has a four-point contact. 
] shall now consider the null-system (Gg), in which to a point G 
are associated the jive straight lines g, which are four-point cuspidal 
tangents for cuspidal curves y" with cusp G. 
14. In each point C of the straight line a I consider the cuspidal 
curve y", which sends its tangent c through P, and determine the 
locus of the points #, which y” has still in common with PC. If 
E lies in P, y" belongs to a system S(; in it (5n—8) curves y” 
occur, which have their cusp on a ($7). So the curve (EZ) passes 
(5n—8) times through P and is of order (6n—11). In each of its 
intersections G with a, a 7” has four points in common with PG. 
The null-eurve of P is therefore of order (6n—11). As it has a 
quintuple point in P, a straight line g passing through P is nul/- 
ray for (6n—16) points G. 
