J 056 



lite point cannot coincide with S^ (v nneqnal to 1), /' is the tangent 

 in *Si to the curve C{p,q) passing through Si- 

 The co-ordinates of the intersection S^ are: 



Cvr{v9—1) Cv(/{vP—l) 



Aiv9 — vi') ' B{v'i — vi') 

 or, by taking into account that v is a root of the equation (2): 



C vP q C v9 p 



A iq—p) ' B iq—p) 



By comparing the equation of the straight line /with the equation 

 of the tangent in the point P^ to Cp{p,q), it is found that /touches 

 only in 07ie point a curve C(/9, 5'). This point of contact is the point /^i : 



Cq ^ Cp 



A(q-p) ' B(q—p) 

 The intersection S^ is the satellite point of the point f\, in which 

 / touches a curve C{p,q) and / and /' are consequently tangents 

 to the same curve C{p,q) in the point P^ and its satellite point aSj. 



§ 5. Of the points in which C{p,q) intersects a straight line / 

 not passing through one of the singular points {() and Y^) of 

 ^\P' Q) 3 ^^'^ ^t most real, if q is odd and 2 at most, if </ is even. 



Let q be odd, and let P^, P^ and P^ be the 3 real intersections 

 of the straight line / with an arbitrary curve C{p,q), their satellite 

 points S,, Ss and S^ lie then according to § 3 on the satellite /' of /. 

 And conversely if the straight line /' intei'sects the curve C{p,q) 

 in 3 points >S,, S^ and S^, the points of which »S,, ^S» and S^ are 

 the satellite points are on a straight line. 



The satellite /" of /' touches the curve C{p, q), which touches 

 /', i.e., the curve C\p,q) which touches / etc, so that we have: 



The straight line /, its satellite /', the satellite /" of /', the satel- 

 lite of the latter and so on, are all tangents to the same curve C{p,q). 



From the similitude of the series of points P^ on / with the 

 series of points /Sj on /' ensues : 



P yP^ '■ PiPz '■ P1P4 = ci-iS^ : >j^C)j : O3O4. 



The straight line /, the three tangents in the intersections of / 

 with the curve C(p,q), the satellite l\ the line passing through the 

 two singular points {0 and Y^) and the tangents in the two singular 

 points touch the same conic. 



If q is even and if the straight line / intersects the curve C(/), ^) 

 in two real points, the tangents in these points to the curve, the 

 straight line /, and the sides of the triangle of co-ordinates touch 

 the same conic. 



