Topological theorems relate in general to the nature, number and arrangement of the dis tinct parts (circuits) of a curve. The first divi sion of circuits, due to von Staudt, is into odd and even, for the number of real intersections with an arbitrary line is always odd or always even. An odd circuit necessarily extends through infinity. Simple examples are the oval and the infinite branch of a bipartite cubic Y-(x-a)(x-b)(x- c)• but circuits may be much more complex. Let the minimum num ber of real intersections with a straight line be called the index of the circuit. Zeuthen showed Ann.,' v. 7, p. 426, 1874) that a quartic may have a circuit of index 2, with two nodes; Cayley showed Mag.,> v. 29, 1865; no. 361, v. 5, Coll. Papers) that a sextic may have a circuit of index 2, without multiple points. It has been proved (Scott, (Trans. Am. Math. Soc.,> v. 3, pp. 388-398, 1902) that for every order m there exists a curve (p =0 or 1), composed of a single circuit of index m-2, or m-4, m-6, down to 0 or 1, according as m is even or odd; any such circuit of index k can be produced by a simple process of linking from k odd circuits. The Zeuthen quartic circuit finds its place in this category; it is due to the linking of two odd circuits; but the non-singu lar sextic circuit is entirely different in char acter, and a general theory of such circuits is still to be suggested.* The possible number of circuits is p + 1. (Harnack, Ann.,' v. 10, pp. 189-198, 1876) ; for every order m there exist curves with this maximum number of circuits, and with every smaller number.
The question of arrangement has been con sidered only with reference to circuits that can be projected into the finite circuits, the so-called ovals. Hilbert ((Math. Ann.,' v. 38, pp. 115 138, 1891) discussed curves with whose simplest type is the annular quartic, com posed of one oval inside another; the number of nested branches cannot exceed “m —2) or }(m— 3), according as m is even or odd; more over, non-singular curves with the maximum number of circuits and the maximum number of nested ovals do exist.
Hilbert draws attention to the arrangement of the 11 ovals of a non-singular sextic (p— 10) ; he states that one of these must lie inside another (p. 118). It appears highly probable (V. Ragsdale, (Bull. Am. Math. Soc.,' v. 11, p. 464, 1905) that this unproved theorem of Hilbert's is the simplest case of a general law in accordance with which at least i(q —1) (q-2) of the circuits of a non-singular curve of order 2q must lie inside some of the remain ing i(q-1)(q —2).
Thus as regards circuits the only question completely solved is that of the possible number; their nature, their arrangement as regards one another, or as regards the straight lines of the plane (on which depends the answer to the inquiry whether all the even circuits can be projected simultaneously into the finite), have been hardly touched upon, though there must be many interesting results awaiting discovery.
Although the general theory of polars be longs properly to the theory of algebraic forms, it supplies convenient expression for some geo metrical facts. The first polar of x, y', z' is the curve of order m— 1, 4=0, where 11 a — a +s asa .
denotestheoperatore—e — A ar second operation with 4 produces the second polar, cf'f==. 0, of order m-2, and so on; thus any point has m —1 polars, of which the last two are the polar conic and polar line. The polar conic of a dp is simply the pair of tan gents; the polar conic of an inflexion consists of the inflexional tangent together with another line which does not pass through the inflexion; thus in both cases the polar conic has a dp. It is easily proved that if the polar conic of B has a dp A, the first polar of A has a dp B. This suggests three derived curves: the locus of points A, the Steinerian, of order 3(m —2)'; the locus of points B, the Hessian, of order 3(m —2) ; the envelope of the line AB, the Cayleyan. Thus, e.g., the curve by which the points of inflexion are determined, the Hessian, is geometrically defined; every point is in a known geometrical relation to f. On the other hand, though it is known that a curve of order (m-2) (ni —9) can be passed through the points of contact of the bitangents, no geometri cal definition of•any such curve has yet been formulated. The curve is of course not unique; what is needed is a geometrical definition of some one curve that shall meet f only at these points of contact, and at the multiple points, in such a manner as to account for the whole number of intersections. (Cayley, 'Coll. Papers,' v. 11, pp. 471, 473).
The metric properties of a curve are im portant in particular questions, though not in the general theory. The curve has m points at infinity; at each of these there is a tangent, which is an asymptote unless it lies entirely at infinity. Consequently for every non-repeated direction to infinity there is an asymptote; for a repeated direction there may or may not be asymptotes. A twofold direction, e.g., may be accounted for either by contact with the straight line at infinity, with no asymptote, or by a dp at infinity, with two asymptotes, dis tinct or coincident. A branch that has a real asymptote is called hyperbolic; a branch that has contact with the straight line at infinity, whether at a single point or at a multiple point, is called parabolic.
The curve has n imaginary tangents from each of the circular points; these by their inter sections determine the n' foci, of which n are real (Pliicker's definition, 1833) ; but the num ber of foci is diminished if the curve passes through the circular points or touches the straight line at infinity.