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15. The system S)) contains o* curves with a triple point 7’. 
If S©) is represented by 
cA HBB 4+7C+6D+ E+ oF =0, 
the locus of the points 7’ is determined by 
Axi Bui Crt Dru Eni Fu) = 0. 
It is therefore a curve (7) of order 6(n—2)’). 
A w with triple point 7’ determines with a nodal dr which has 
its node in 7, a pencil of nodal dr with fixed tangents d,d’. The 
net of the curves d” with node: 7’ therefore consists of o' similar 
pencils of which the tangents d,d’ form an involution. Each of the 
two nodal rays c,,c, is common cuspidal tangent for a pencil of 
cuspidal curves and each of these two pencils contains a y” with 
four-point tangent. The five null-rays g of 7'are therefore represented 
by the straight lines c,,c,, and the three tangents ¢,,¢,,¢, of the 
curve vt”. The points 7’ are consequently not singular. 
16. In a sextuple infinite system SS each point 7’ is triple point 
of at. To 7 as null-point the three tangents /,, ¢,,¢, of t” are now 
associated as null-rays. 
In order to find the second characteristic number of this nul/- 
system, 1 consider the curves 1?, of which the point 7’ lies on the 
straight line a and | try to find the order of the curve, which 
contains the groups of (n—3) points 4, in which ¢ is moreover 
intersected by PT’. 
If # lies in P, t belongs to an S®), and 7’ is one of the 6(n —32) 
points which ($ 15) the curve (7’) has in common with a. So F is 
a (6n—12)-fold point on the curve (£), which consequently has the 
order (7n—15). 
The nudl-curve of P is therefore of order (7n—15). As it passes 
three times through P, a straight line ¢ passing through Pis tangent 
for (7m—18) curves t+”, which have their triple point 7 on t. A 
null-ray, therefore, has (7n—18) null points. 
17. The curves (7), which belong to two systems S©) comprised 
in S®, have the 15(2—2)* points 7 of the system S® in common, 
which forms the “intersection” of the two S©), 
The remaining intersections are critical points, viz. each of them 
is triple point for a pencil of curves t”, consequently singular null- 
point S for (7,4. This null system has consequently 21(n—2)' 
singular null-points. 
1 If, for n—=3, the system S(5) has the base points B, By, Bs, By, the curve 
(T) consists of the straight lines Be Bi. 
