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As D represents two coinciding base points, there is a c*, which 
has D as node; the locus of the points D is the curve of Jacobi 
of the net, A‘. 
The 18 tangents ¢ are at the same time tangenis out of P to the 
curve (/)'", determined by the null system sg» and possessing an 
octuple point in P. 
The curves (P)' and A* have besides the 18 points of contact 
of the straight lines 7, moreover 18 points D* in common. Evidently 
PD* is one of the tangents d,d' in D* at the c?, which bas D* as 
node. The nodal tangents of the rational curves of the net envelop 
therefore a curve of the 18" class *) (curve of ZrUTHEN). 
The pairs of tangents d,/’ determine on a straight line / a 
symmetrical correspondence [18], which has double coincidences in 
the 6 points D lying on /. The remaining coincidences arise from 
tangents in cusps; the net, therefore, possesses 24 curves with a 
cusp. . 
Let us moreover consider the correspondence (36, 18), which is 
determined on / by the straight lines ¢ and d. Here too the 6 points 
D lying on / are double coincidences; the remaining 42 arise from 
straight lines ¢, which have coincided with one of the nodal tangents d. 
In the corresponding point D=— 4° the curves of a pencil (c°) have. 
evidently three coincided points in common. 
The net consequently contains 42 pencils, the curves of which 
osculate each other. 
$ 3. If a point / is made to describe the straight line p, its null 
rays envelop a curve (p)' of- class six, which has p as triple 
tangent. The two remaining null points of 7 will then describe a 
curve a, the order of which we can determine by investigating 
how many points it has in common with p. To them belong in the 
first place the 6 points D lying on p, for on the base tangent ¢ 
belonging to D, the point D represents two points /. a has further 
evidently nodes in each of the three null points of p; it is 
consequently of order 12. 
the triple points are found from 
wi 3d 1+ Jhr = 0; 
the neutral points from 
©, + a(e,+2,)+ 6=0 and aa,e, + ble, +) = 0. 
They coincide if 5 =0; but in that case two triple points have coincided in « = 0. 
1) If to each point D the two tangents d,d’ are associated a correspondence 
(1,2) arises between the points of Af and the tangents of (d)!5. From the formula 
of correspondence of ZeurHeNn we find then that (d)!* is of genus 31. 
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