1059 



contact for the arbitrary point " L, it has been again proved 

 that the points of contact lie on the conic k[,. As the equation of ^'L 

 does not contain the constant c, which determines the curve C{p,q) 

 the points of contact of the tangents from L to the curves of 

 the pencil lie on the conic ki. 



If q =z3, the first polar is of the second order and passes 

 through the cusp of the cubic. The intersections of the first polar 

 and ki are the cusp and the three points of contact. 



§ 9. The tangent at the point P{ati>,bt'i) to the curve Cp{p,q) 

 has the equation : 



?f._ py_— _ 



ntP bt1~~ 

 The pole of this tangent with regard to the conic / : 



?•«' — py' = q — P» 



is the point R\ — , — I . 

 \ati> bty 



If P describes the curve C(p,q), the locus of R is a curve 

 C{p,q) too. The reciprocal polars of the curves of the pencil 

 C{p,q) with regard to the conic ƒ are therefore curves of the 

 same pencil C{p,q). The straight lines through L{l,n) are the 

 reciprocal polars of the points of a straight line /. The points of 

 contact of the tangents from L to the curves C{p,q), are the 

 reciprocal polars of the tangents at the points of / to the curves 

 C{p,'q) and as these points of contact lie on the conic kj^, the 

 tangents in . the points of a straight line envelop a conic ki, 

 as was formerly proved. And as Icl passes through the 

 vertices of the triangle of co-ordinates that is autopolar 

 for /, ki touches the sides of the triangle of co-ordinates. The 

 conic ki passes through the point L and through the point L^, 

 which has L as satellite point. The conic ki touches therefore 

 the polar lines / and /, of L and L^. As L and L^ lie on the same 

 curve C (j), q) I and l^ touch the same curve C[p, q), and as L 

 lies on the tangent in Lj, the point of contact of /^ lies on /aud/^ 

 is consequently the satellite of /. 



As the straight line /, if q is odd, intersects a curve C{p,q) in 

 three real points at most, and if q is even in two real points at 

 most so, if q is odd, three of the tangents from L to a curve 

 C{p,q) are at most real, and only two at most, if q is even. 



^ 10. Let q be odd, and P,(iv,7,) a point of the curve q of 

 order 7i: 



