1058 



passes through a vertex of tlie triangle of co-ordinates the tangents 

 to the curves C{p,q) in the points of the straight line pass through 

 one point. 



If (p is a curve C{p,q) this curve is the envelop of those tangents, 

 and as C{p,q) passes p times through 0, and q — p times 

 through F^ and does not touch k^in these points, 2(/ — p — {q — p)=^q 

 intersections fall outside the vertices of the triangle of co-ordinates, 

 and the curve C{p, q) is therefore of class q. 



The parameters t of the points Q, in which the curve C{p, q) 

 passing through P{a,b) intersects the conic k/^ , i.e. the parameters 

 of the points of contact of the tangents out of L {X, ^) to Cp{p,q) 

 are the roots of the equation: 



atf' htl ^ ^ ■ 



1 

 This equation is in - of order q, and it is easy to see that, for 



q odd three of the roots at most, for q even two of the roots at 

 most, are real. 



Consequently we have : 



If q is odd three real tangents may at most be drawn out of an 

 arbitrary point to a curve C{p,q). The three points of contact of 

 these tangents lie on a conic passing through the vertices of the 

 triangle of co-ordinates, through L and through L^, in which L^ is 

 the point of Ca {p, q) that has L as satellite point. 



If q is even, only two real tangents may be drawn out of an 

 arbitrary point L to a curve C{p,q). The points of contact of these 

 two tangents lie on a conic passing through the vertices of the 

 triangle of co-ordinates and through the point L. 



§ 8. The equation of a curve C{p,q) is: 



yP = c xt. 

 The tirst polar of a point L {X, n) is: 



Xcqx9-^ p n ijP—^ — (q —p) ijP == , 



Consequently the points of contact lie on the curve : 



x\Xcq A-9— 1 — pix y^'-i — {q —p) ijl' \ — q).{c x'i — yP) — 0, 



or after reduction 



.V^-i { {q—p) xy -\- Pit x—q Xy\—0, 

 SO that the points of contact lie on the .T-axis, the line at 

 infinity, and on the conic kj/, and as the line at infinity 

 and the .r-axis intersect the curve C{p, q) each q times in 

 one of the singular points, which singular points are no points of 



