( 432 ) 



Rrs two movable points of intersection of Cr and Cs, coincide so that 

 the point Ajs as a point of the curve of contact is found when Cr 

 and Cs show in Ars a contact of the second order which takes place 

 three times. Further Rrs passes through the doublepoints of the curves 

 Cr and Cs, of which the number for the pencil {Cr) amounts to 

 3(r — 1)^ and for the pencil {Cs) to 3(5 — 1)', which follows imme- 

 diately from the order of the discriminant. 



Each of the r* — ^' — y' ■ — ö points Ar is a simple point of inter- 

 section of Rrs fïnd Rrt (simple, the tangents in Ar to Rrs and Rrt 

 being the tangents of the curves Cs and Ct passing through Ar, 

 diifering thus in general), but no point of Rst ■ Each of the «' points 

 Ast is a double point of intersection of Rrs and Rrt , as those curves 

 of contact in Ast have a simple point with the same tangent, namely 

 that of the Cr passing through Ast ; these points are also points of Rst , 

 namely threefold ones. Each of the /?' points Art is threefold pomt 

 of inlersection of Rrs and Rrt (it being simple point of Rrs and 

 threefold point of Rrt ) and lies at the same time on Rst ; the same 

 holds for the y' points .4,* . Each of the ff points Arst which are common 

 basepoints of the three pencils is 9-fold point of intersection of Rrs 

 and Rrt , being threefold point of each of those curves ; moreover it 

 is threefold point of Rst- B'inally the 3(?" — 1)" doublepoints of the 

 pencil {Cr) are simple points of intersection of Rrs and Rrt , but not 

 points of Rst ; of the curves Cr, Cg and Ct passing through such a 

 doublepoint Cr has an improper contact with Cs and with Ct, without 

 however Cs and Ct touching each other. 



From this we see that the curves of contact Rrs and Rrt have 

 ,,» _ ^' _ y' _ (f ^ 3 (/• — 1)'' = 4r» — 6r + 3 — ^' — r' — d 



points of intersection which are not points of Rsi , and so do not 

 furnish coinciding points ]\ P' . Moreover Rrs and Rrt have 



2«' + 3ii' -4- 3y' -I- 9J 

 points of intersection coinciding with the common basepoints, which 

 do fall on Rgt , but which do not give any coinciding points 1* and 

 P', as for this it is necessary that of three curves C,, C^^ and Ct 

 passing through the same point each pair shows two movable points 

 of intersection coinciding with that point. So for the number of coin- 

 ciding points F and /-*' remains : 



^2r 4- 2s — 8) (2r -f- 2« - 8) — (4r' — 6/- + 3 — |J' — y' — d) — 

 — (2«' -f 3/3' + 3y' + 9(f) = 



= 4(.sr-|- tr -h rs) — 6{r -f s + f) + 6 — 2 («' + /J' -f y' -f 4d). 



So we find : 

 It happens 



4(,^ + tr f rs) — 6(/' + s -\- t) -]- Q — 2(« + /? + y + d) 



