These six are Pliicicer's equations; three inly are independent. It must not however be aipposed that we can choose any three of the lumbers arbitrarily. There is a limit to the lumber of dps that a proper curve can have; + K > i On 1) (15 2). For if the curve lad one more, a curve of order an 2 could be passed through the dps and m-3 other points in the curve; but this would have with f inter sections in number n-3; i.e., m(m 2)1- 1, which is impossible. k curve can actually have this number of dps; t is then called rational (or unicursal), be ;ause the co-ordinates of any point can be ex iressed rationally in terms of a parameter, as Follows: The dps and m-3 other points on f determine (m 2)-ics with one degree of freedom, a system u + kv =O. Of the ns (m 2) intersections of one of these with f, all except one are at the fixed points , the :o-ordinates of the one remaining variable in tersection are therefore given rationally in terms of the parameter k by f 0, u + kv =--- a If the curve has not this number of dps, let it have (i X); the number i(m 1) (m-2) (5, which has already been given as the genus, was originally called the deficiency of the curve (Cayley).* From Pliicker's equations we find (In I) (m 2)v K-=i 1) i.e., the point deficiency and the line deficiency arc equal; the deficiency (or genus) does not belong specially to the point system or the line system. In particular, if the curve is rational qui locus, it is rational qui envelope. If we introduce p for the genus, the equations assume a convenient form, n+x--2m+ 2(p-1), m-1-&---2n+2(p-1), m (m-3) 2 (v+K)=----2 (p =n (n-3)-2( r + t).
There are other limitations on these num bers, but the theory is not complete. Clebsch proved (Crelle, v. 64, p. 51; 1864) that the num ber of cusps on a rational curve cannot exceed (m 2) ; the more general question as to the maximum number of cusps for a curve of assigned order awaits solution.
So far the singularities have been supposed to be simple. A multiple point of order higher than the second is, in a certain sense, equivalent to a number of simple singularities. If the tangents at a k-point are distinct, the point arises from the crossing of k branches; these cause ik (k 1) intersections, nodes by the geometrical definition. It can be shown that such a point reduces the class of the curve, the number of inflexions, and the genus, by 2 X ik (k 1), 6 X ik (k 1), ik (k 1) respec tively. Thus not only is the _point explained geometrically by ik (k 1) nodes, but as re gards Plficker's equations it is equivalent to these nodes. Nevertheless the point cannot be replaced by these nodes for all purposes, e.g., as regards the number of conditions; the k-point imposes ik (k 1) conditions, whereas the nodes would impose I k (k 1), i.e., (k 1) k (k 1). It is an important fact that this equivalence does hold as regards the conditions imposed on adjoint curves; the pres ence of the (k 1)-point on the adjoint im poses ik(k 1) conditions, equal to the num ber that would be imposed on the adjoint by ik (k 1) separate nodes on f.
If the tangents are not distinct, the matter may become very complicated. The multiple point immediately revealed by the equation may be but one of a series of multiple points indefi nitely close together, and the singularity then involves also multiple tangents. The determi nation of the point- and line-components, the ((analysis of higher singularities,') has received much attention. If the singularities are re garded as singularities of the equation, the question is properly considered from the alge braic standpoint. At an ordinary point of the curve, y can be expanded in an ascending series of positive integral powers of x, provided x=0 is not the tangent; at a k-point the process is not directly applicable, it requires some modifi cations, and then leads to k expansions, with exponents either integral or fractional. An ex pansion with fractional exponents, whose L.C.D. is q, is accompanied by the q 1 conjugate expansions, thus forming a Cycle of order q (Puiseux). The integral expansion is a cycle, q= 1, and the k-point is represented by a num ber of cycles of orders qi, q,, qa, . . . where k. The number of cusps in a cycle is q 1; this agrees with the known facts about the simple cusp y= and is ac cepted as the algebraic definition of cusps. The algebraic definition of a node is indirect; v is determined so that 2v + 3K shall be equal to twice the total number of intersections of all the branches (the discriminantal index of the singularity) ; it is proved that this definition yields always a positive integral value for v. (Chrystal, v. 2, 1889, pp. 359-371; Harkness and Morley, 'Treatise on the Theory of Functions,' 1893, pp. 127-151; Brill-Noether Bericht, pp. 367-402 for full references; (Die Entwicklung der Theorie der algebraischer Punctionen in alterer and neuerer Zeit' 1894).
The process as outlined above, dealing with the expansions as a whole, simply enumerates the components, algebraically defined, of a sin it affords no clue to the structure. This structure can however be put in evidence by means of the critical exponents of the ex pansions, those in whose denominators a new factor appears. These show that the dps are combined into certain multiple points, and thus they lead to an algebraic description of the sin gularity. (Smith, H. J. S., (Proc. Lond. Math. Soc.,' v. 6, pp. 153-182; 1876; Halphen, pros. a l'Ac. des Sc. de Paris,' v. 26, 1877; Zeuthen, Ann.,' v. 10, pp. 210-220, 1876). A different treatment of the expansions, due to Noether, gives a clearer idea of the structure.