Lectures on the Theory of Plane Curves

The ratios of the distances- of the three fixed points from the line, measured in a fixed direction, are sufficient to determine the position of the line uniquely. [...] If through the vertex 0 a plane is drawn ►aralled to p cutting the plane p' in a line then s' is the projection on p' of the line at infinity of p. Similarly if the plane, drawn parallel to p', intersects p in a line s, 8 is the pro- jection of the line at infinity of p'. [...] If m=n, the above theorem becomes—if of the ng points of intersection of two curves of the nth degree, n 2 — 4(n— 1)(n — n(n + 3)-1 are given, the This theorem also holds if the curve of the nth degree breaks up into two curves of the kth and n —k ill degrees respectively, when the curve of the mth degree does not puss through all the intersections of the two enrves—Seo Zeutheu—Sur to determinati [...] For supix)se one ml►re of the points lies on the curve of the mth degree ; in that ease the curve of the nth degree would meet the curve of the ,nth degree not. [...] These considerations led to the development of a theory of groups of points on a curve, the principles of which are contained in a paper by Noether. * The principles of the theory are illustrated in the following theorem :- If a group of n (/÷m) points on a curve C. of order is forms the complete intersection of C. with a curve Noother—" Uuber einon Bata anti der Thoorio dor Algebraianhen Futtotio