294 



B^,B*; let this base-point be indicated hy A^, while the sixth base- 

 point will be indicated bj 6'. 



Now [c*] contains a threeside formed by a^^A^B^B^* a^ii^A^BtB* 

 and a stiaight line a^, which contains C and forms with y' tiie 

 curve {By of C. The singular .straight line a^ bears an /,, of which 

 the pairs are completed into groups of the /, by A^. 



To a^ belongs again fas in § 3) a pencil of conies, the curves of 

 which are completed by a^ into figures c'. This (c') has as base- 

 points Bz, B^*, C and a point J^,. which is singular, because it 

 forms groups of /, with the pairs of the /,, which (c*) produces 

 b}' the intersection with fi^. Amilogously there is Si singalar point A^ 

 to which an /, l)elongs placed on a^. 



To the pencil (c*), which is associated to a^, belongs the figure 

 formed by a^^ B^B,*, and the straight line CA,; the latter is 

 therefore identical with the third straight line a, of the threeside 

 mentioned above. Analogously a■^ and a, form one of the conies 

 that are associated to «,. P'rom this we conclude that the singiilar 

 points (',.l, and .1, are collinear, and lie on the siiKpdar straight 

 line a^. 



To the peiu'il associated to a^ belongs also the pair of lines (7ij, 

 A^^B,*; on the second of these lines the net detei'mines an /' or 

 pairs (A^ A'), which are each completed into triplets by a point of 

 CB^. So the lines A^B^*, A^B,, A^B^^ and A^B.^^ are singular; we 

 may indicate them by 6,*, b^, i^*,, b^. 



Finally there is moreover a singular straight line c, which passes 

 through C and forms with the threeside a^a^a^ the curve (7^)'' of 

 C. It contains an I^ of pairs A', A', which are every time base- 

 points of pencils out of [c'j. if we now take two ai'bitrary fixed 

 points M and M', and if we associate the two c\ which each of 

 the pencils in question sends through M and M', two (c") are on 

 account of this made projective. As any two homologous c* intersect 

 each other in three points of c, and the two pencils have a curve 

 c% in common, the figure |)roduced by them consists of c/, the 

 line c, and a conic y'; the latter therefore is the locus of the point AT". 



Summarizing we find that this /j, has eight singnlar points and 

 eight singular straight lines. 



Its coincidences lie on a y\ which passes through the points B 

 and twice through C. 



In an analogous way, as in § 3, it appears that A envelops a 

 curve of the third class, when X describes a straight line. 



5. Let us now suppose that the base-points B^, Z>,, /?, are respec- 



