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witli A'. From this it ensues that the rajs A' A" envelop a curve 

 of the third class, wliich evidently has C\ as bitangent. The 

 straight lines .6'= A' A'", which are indicated by the triplets of the 

 /,, form the triplets of an invobition of rays /j. For this involution 

 too, Ck is singidar, as it belongs to oo^ groups; the straight lines 

 x^x" form an i„ for which 7^■' is the curve of involution. 



2. When a point A' describes the straight line p, the rays .v', x", 

 which connect A' with A", A'', envelop a curve {p)^ of the fourth 

 class, which lias /> as bitangent. The curves {p)^ and {q)^ have 

 16 tangents in common; to them belong the rays .c',.?;", which emanate 

 from X = p(/, and the six singular sti'aight lines f^.. There are conse- 

 quently 8 straiglit hues x", for which A' lies on /;, and A^' on q. 

 In other words, if A describes the straight line /;, A' and A" describe 

 a curve />^ The latter intersects /) in the first place in the pair of 

 the /, lying on p, and further in six points A', which have each 

 coincided with a point A", consequently are coincidences of the /g. 

 The coincidences of the /, fonn therefore a curve of the sixth 

 order, y\ 



If two base-points of a [)encil meet in a point B, there is a curve 

 that has a nodal point in B. So y" is at the same time curve of 

 Jacori for the net [c'], has consetpiently nodal points in the six 

 base-points Ck- In each of these |)oints it has the tangents in common 

 with the nodal curve y^k. Outside the points C the lines y" and y'jfc 

 have only two more points in common; they are the coincidences 

 of the involution (A', A") lying on fk- 



The curve (/>) is of order 10, is therefore cut by p in 6 points. 

 For each of these intersections A, .//' coincides with .r', consequently 

 A'' with X" . The locus of the "bnrnch points', the ''complementary 

 carve" is consequently also a curve of the sixth order, x^. It has 

 nodal points in the singnlar points Ck, because y// bears two coin- 

 cidences. The curves y' and .i-" have besides the 6 points C more- 

 over 12 points m common, they are united in pairs into triple points 

 of the /j. So there are in /, six groups, in lohich the three points 

 are united in one point. 



The above mentioned curve />** has a triple point in Ck, because 

 y/,' has three points A in common with p, for which A" lies every 

 time in Ck. 



The pairs of the /,, which are collinear with an arbitrary point P, 

 lie on a curve {P)\ which passes twice through P and contains the 

 singular points 6"). So />' and {Py have in Ca; 18 points in common; 



1) For Ck this curve consists of yk'^ and the straight line Ck. 



