The fundamental theorem is the Cayley Bacharach theorem (1843, 1886) a C through all except 11(1+ m n 1) (i+ m n-2) of the intersections of Cl and C., will pass through the remainder unless these lie on a curve of order 1+ m n 3. In particular if precisely }(m 1)(m-2) of the intersections of a fixed C, by a variable C. are a consequence of the rest if n> m-3; hut if m-3, fewer than }(m 1)(m 2) are a consequence. Thus no matter what the order of the cutting curve, the number of the in duced intersections can never exceed i(m (m 2). This number, which limits the inter connection of points on the curve of order in, is called the genus of the curve, and is usually denoted by p. The genus of a quartic, e. g., is 3; 3 points on a quartic follow from 5 intersec tions with a conic, or from 9, 13, 17 intersec tions with a cubic, quartic or quintic, and so on.
If a curve has multiple points, corresponding theorems hold and preserve their significance, provided the cutting curve is an adjoint curve, that is, a curve with a multiple point of order k 1 where the fixed curve has a multiple point of order k. The genus, the number p which limits the interconnection, is in this case ( m 1) (m 2) ilk (k 1). (Macaulay, 'Proc. Lond. Math. Soc.,' v. 26, pp. 495-544; 1895; Hardcastle, F., 'Report on Point-Groups,' in progress in Proc. of the Brit. Assoc.; Brill Noether Bericht, Entwicklung der Theorie der algebraischen Functionen in alterer and neuerer Zeit' (1894).
Multiple points arise when two or more points of the system occupy the same position in the plane; according to the number of points that coincide, the point is double, triple, etc. Double points (dps) are nodes or cusps; a node is the coincidence of two non-consecutive points, a cusp is the coincidence of two consecu tive points; similarly double tangents are either bitangents or inflexional tangents. The cusp and inflexional tangent are also called sta tionary point and line, on account of the effect of the singularity on the motion of the point and line by which the curve is described.
The coincidence of two points at P causes an arbitrary line through P to meet the curve in two points there; hence if P be taken as origin, the terms of the first degree in the equation will vanish. The conditions for a double point at x, y, z are therefore the vanish ing of ax af af ( in homogeneous co-ordinates, a of af , ay . Two lines can be found to meet ax s the curve in more than two points at P; these are the tangents; they are given by the terms of degree 2 equated to zero. Similarly the condi tions for a k-point are the vanishing of the k__i }k (k + 1) derivates a (a+ Oxaelydosr and there are k tangents, lines which meet the curve in more than k points.
If the tangents at a dp are distinct, the point is a node, formed by the crossing of two simple branches, real or imaginary accord ing as the tangents are real or imaginary. In this last case, the point is detached from the main body of the curve, and is called an isolated point, or acnode, the visible crossing being called a node, or crunode. If the tangents are coincident, and meet the curve in precisely three points, the point is a cuspi if the coin cident tangents meet the curve in more than three points, the singularity is complex.
a a f aY as Elimination of x: y f from = 0 =--0. a Ox z gives a condition, D=0, to be satisfied by the equation of any curve that has a dp. Hence a general locus of order m has no point singularities, and a general envelope of class n has no line-singularities. But a general en velope has point-singularities and a general locus has line-singularities; for the direction of a tangent, given by the value e Of is un dx changed if 0, i.e., if 2 _ 2 ay of /ant = ax= ay kcTici This condition, and Pc=0, are only two equa tions, and consequently no condition is imposed on the locus by the existence of line-singulari ties.
If a curve of order m, class n, has v nodes, g cusps, r bitangents, 1 inflexional tangents, these numbers are connected by Pliicker's equations. The tangent at x, y, z is ...,a.f . .03 . .
A -r =---- V; &X OY OS this passes through x, y, z if x, y, z lie on as f the curve x' + y' + a' =0, thepolar of x', as y', z' with respect to f. Since this curve, di °°. 0, is of order ns 1, it meets f in m(nt 1) points; two of these lie at every node, three at every cusp; hence the number of tangents from x, y', z' is 3z. . . (1) Similarly from the line-equation, m=n(n-1) 2r 31. . . (1') The condition that the tangent at .r, y, z be stationary becomes in homogeneous co-ordinates r , 0; 1 ax ay ay ax ay ax oaf aif alf al f asf ax as ay as ass the point of contact must lie on this curve, 1(f)= 0, the Hessian of f, of order 3(nt 2). At a node on f, H has a node with the same tangents; at a cusp, H has a triple point, com posed of two branches touching the cuspidal argent with one branch cutting it; the nuttiL Hers of intersections are 6 and 8. Hence & -=-. 3m (m 2) 6v 8s, . . . (2) and reciprocally k =-- 3n(n-2) 6r 84. . . . (21 Any one of these four equations can be ob ained algebraically from the others. From hese we can find also (3 and 3') expressions or r in terms of m, v, z, and for v in terms of 1, r, I.