109 



a pair of the /•\. then (he lines joining B^ to tlie two points, deter- 

 mined by the f/% which touches </^ in B^. The intersections of / 

 and (/^ procure three common tangents of {x)^ and (.j;),, ; tliere are 

 consequently 15 straight lines, whicii bear a pair Y', Y" , so that 

 the said correspondence associates 15 curves ^/' to 7^ 



By means of this correspondence the points of a straight line /• 

 are arranged into a correspondence (30,30). For to the (('^ passing 

 through a point R of r cori-espond the 30 intersections A'' of /-with 

 the 10 curves f/^ associated to f/* ; on the other hand the r/? passing 

 through R' procures 30 points R, by means of the corresponding 

 15 op^ The intersections of the coiTcsponding curves form therefore 

 a figure of order 60 ; it consists, however, of two parts : the locus 

 of the pairs Y' , Y", which lie on the tangents of the {,v)^, and the 

 locus of tlie points Y. 



The former may also be produced by the pencil {jf") and the 

 system of rays {.i)^. To each 7'', in virtue of the consideration men- 

 tioned above, a number of ten straiglit lines is associated, which are 

 each coupled to one 7^ oidy ; hence a (10,12) arises now on r, so 

 that the ])airs of |toiuts )'^ Y" are lying on a figure of order 22. 



For the points Y we tind therefore a figure of order 38 ; it is 

 composed of the three lines h,,,,, and a ciu-ve of order 35. For to 

 the iiitersection A^ of / and 7>, /x^ corresponds a pair A', A" on /I />.^; 

 but this line bears 00^ pairs Y' , )" and the corresponding points Y 

 of B^B^ are all associated to X. Apart from these three lines the 

 line / is transformed by means of .the birational correspondence 

 (A',]") into a curve of order 35, /."". It cuts / in 10 pairs A', !'(§ 4) 

 and in 15 coincidences A' = Y. There is consequently a cuirc of 

 coincidences of order fifteen. The figure of order 22 found above 

 consists of the three lines hk and a curve /'", for to the conic (63, Z^j,) 

 corresponds the tangent b^ of {.v)^. 



8. We shall now determine the fiuulamentnl curves which are 

 associated to the fundamental points A, B^, Cu. The curves of invo- 

 lution (.iOs belonging to |i/ and ,5./ (§ 2) have 9 tangents in common, 

 there are consequently 9 lines, for which A lies in i>, and 3" in B^. 

 Therefore the fundauiental curve of 7i, has nonuple points in B..^ and 

 ^3. No other point Y of the line B^B^ can correspond to a point 

 A' lying iu ./ii ; the said curve is therefore of order 18, It has a 

 nonuple point in ]j too and passes three times through each of the 

 points A and Ci, ; for through T or J//, passes one line, bearing a 

 pair A', A" of ,-?i'* and a pair Y',Y" of n^ or y/,"' ; tiirough whicli 

 then Z>i = A corresponds to a point Y lying in A or 6/,. 



