On Plane Curves of Degree n with a Multiple Point of Order n −1 and a Conic of 2n -Point Contact

Harold Hilton
1922 Proceedings of the London Mathematical Society  
1. We have considered elsewhere the properties of a plane algebraic curve of degree n (an n-ic) with tangents of n-point contact {Messenger of Mathematics, 1920). The case of an ;i-ic with a conic or conies of 2?tpoint contact at once suggests itself. It will be found immediately that an w-ic meeting y = x 2 in 2?i-points coinciding with the origin (which is not a double point) has an equation of the form (y-x 2 )u n -2 = y 11 , where u n _ 2 = 0 is some {n-2)-ic. Then, by a change of axes, we
more » ... change of axes, we have the result that an n-ic meeting the conic u 2 = 0 2n times at the point of contact of its tangent u^ = 0 has the equation 2. In this paper we confine ourselves to the case of an w-ic with a multiple point B of order n-1 [an (n-l)-ple point] meeting a conic 2 2n times at C. Let the polar of B with respect to 2 meet the tangent at C in A, and meet BG in 0. Let BC meet 2 again at D. There are two cases to consider. In § § 3-8 we discuss the case in which B is outside 2, and in § 9 the case in which B is inside 2. 3. Let B be outside 2. We shall find the following notation useful later on. Let U, V be points on BG conjugate with respect to 2, and such that the cross-ratio UB. CD/UD. CB of the range (UBGD) is 2/fa+l), and therefore the cross-ratio of (VBGD) is 2rc/(n+l). Let AO meet 2 in E, F, and let A V meet the tangents BE, BF from JB to 2 in I, J. Let W, X be the points on BG such that the cross-ratio of (WBCD) is 8rc/(2w+l) 2 , and the cross-ratio of (XBCD) is 4M/(2-»+1). Let (BO, XY) be a harmonic range. It is possible to project 2 into the circle x 2 -\-y' 2 = y, A being the
doi:10.1112/plms/s2-20.1.93 fatcat:e2kja2meg5helazk5rbpqep2ai