295 



lively lA'iiig on the sides A, A^, A^ A,, A, A, of the triangle which 

 has the base-points Ak as vertices. 



In the same way as with the preceding /,, there belongs to the 

 straight line d,* ^. A^AJ^, a (c'), the base of which consists of 

 A,, B^, /i, and a certain point- /I,*, which is again sim/?i//(r and 

 belongs to an /, lying on a*. Analogously there are moreover two 

 other sini/ular points, A/' and A^, which are related to involutions 

 /, on a^* and a*, it appears now from the consideralion of the 

 threeside formed by a,*,a^* and by the straight line '/„, which must 

 pass through B^, that a^ contains the points A* and .1*, (see § 4). 



Evidently the nrujular straight lines (t,,(f.„/i, belong respectively 

 to the singular points A^, A^, A^. 



The ?iine singubir points are now placed in such a way, that 

 each point B^ is the intersection of the lines a^. and ci/,* ; the triangles 

 A^A^A, and A*A^*At* are consequently circumscribed to the points 

 B^,B,,B,. 



Besides the six singular straight lines ((/.■, "k* theie are moreover 

 three singular straight lines bk^^ AkAjc*- For, on the pair of lines 

 A^Aj*, B^Bf [c^J determines groups of the /j, of which every time 

 one point lies on B^B^, while the other two form a pair on h^. 



The curve of coincidence is now a y\ which passes through the 

 points B. To a straight line p a // is associated, while the straight 

 line A' envelops a curve of the second class, when A' describes the 

 straight line /;. 



For B, the curve {By consists of a /?,' {A,B,B,B,A,*) and the 

 lines a^,a^*; for each of the remaining singular points it consists of 

 four lines easily to be indicated. 



For further particulars I refer to my paper mentioned above. 



6. We now considei- a net [c'] that has the vertices .1^/ of a 

 fourside, with sides ai-, as base-points. To the straight line a^ a 

 (c') is associated, which has as base-points ^,^j, J^^, Jj^ and a certain 

 point ..4,; each of these c"^ forms with a^ a figure of the net. To 

 these figures belongs the threeside that is composed of a^, a^ and a 

 third straight line a^^, which must pass through A^^, but cannot but 

 contain the singular point A^. But this threeside may at the same 

 time be considered as compound of the straight line a^ with a pair 

 of lines of the (c"), which has as base-points A^^, A^^, A^^ and a 

 certain point A^\ consequently the third straight line a,, passes 

 through J,, and .4,. The singular straight line («i., contains therefore 

 the three singular points A^, A^, Af^. 



Besides the six singular points .4^-/, which have each a straight 



