manifest that if a curve is to touch a given straight line at a given point, this circumstance is generally equivalent to two conditions in the determination of the constants of the curve. See Art. 211. Thus as five conditions will determine a central conic, it is generally impossible to construct one which shall touch the three sides of a triangle at given points. As four conditions determine a parabola, so a parabola is determinate if it touches two given straight lines at given points.

223.] Now (§, n) in (52) is any point on that tangent which touches the curve at (x, y); let us suppose (§, 7) to be a fixed point, and tangents to be drawn from it to the curve; then as (52) is of n-1 dimensions in terms of a and y, all the points of contact lie in a curve of the (n-1)th degree; and therefore there may be as many points of contact, and consequently as many tangents as there are points of intersection of (52) with the original curve: that is, there may be n (n − 1) tangents drawn from a given point to a curve of the nth degree. If the n (n-1) roots of the equation which determines the points of intersection are all real, they give points at which tangents may actually be drawn to the curve from (, ); but the tangents which correspond to imaginary roots cannot be drawn, at least to those branches of the curve which are in the plane of reference. Hence the theorem gives the number which the tangents drawn from a given point and in the plane of reference cannot exceed. Since n (n-1) is an even number, all the roots may be imaginary, entering as pairs of conjugate roots, and therefore it is possible that no tangents can be drawn to a given curve from points within certain portions of space in the plane of reference: for and enter into the equation (52), and therefore the reality of the roots of the equation which determines the points of contact is dependent on their values. This is geometrically evident because from all points towards which a curve is concave no tangents can be drawn to that relatively concave part whereas for all those points towards which a curve is convex two tangents can be drawn to that relatively convex part. And if the point whence the tangents are drawn varies, the two tangents become coincident when it is on the curve. And thus from a point on the curve only n (n-1)-2 tangents can be drawn to the curve. This is in accordance with the algebraical theorem, that if a pair of conjugate roots of an

equation gradually varies, and becomes real by reason of the vanishing of the imaginary part, they are equal at the values where the transition from their being imaginary to their being real takes place.

Although the tangents corresponding to the imaginary roots may not be capable of geometrical exhibition, yet as they are important in reference to a subject which we shall presently investigate, their algebraical existence must not be forgotten. These theorems are also true when the point from which the tangents are drawn is at an infinite distance; that is, in other words, when the tangents are all parallel to each other.

If n = 2, the original curve is a conic; and n(n-1) = 2; therefore not more than two tangents can be drawn from the same point to a conic; and the point whence the tangents are drawn may have such a position that two may be drawn, or only one, or none: thus as to an ellipse; from all points outside the curve two real tangents can be drawn: from all points on the curve, only one: and from points within the curve, none.

If n = 3, n(n-1) = 6; and therefore six is the greatest number of tangents that can be drawn from a given point to a curve of the third degree; and if the point from which the tangents are drawn is on the curve, only four tangents can be drawn. We shall hereafter have various modifications of these propositions.

Plane curves are also arranged according to the number of tangents which can be drawn to them from a given point: and according to this character they are said to be of a certain class. Thus if a curve is capable of having m tangents drawn to it from a given point, it is said to be of the mth class. Thus, if a curve is of the nth degree, it is of the n (n-1)th class. A curve of the second degree is of the second class; a curve of the third degree is of the sixth class.

It is also manifest by general reasoning that not more than n(n-1) tangents can be drawn to a curve of the nth degree from a point in its plane. A tangent meets a curve in two coincident points; and as a straight line drawn from a given point cannot cut a curve in more than n points, so the number of pairs of points of intersection of the line with the curve may n (n-1) ; and when two points of intersection become coin

be

2

cident, the cutting line becomes a tangent; and as each pair of points may become coincident in two different ways, so the number of tangents may be n (n−1.)

224.] Here we must investigate another method of finding the equations of the n (n-1) tangents to a plane curve, which will give the equations of all in a synthetic form; and the process too being general will lead us on the way to other more general theorems of algebraic curves. For the sake of symmetry and homogeneity I shall use the equation to the curve in the triliteral form, and shall express it by the equation,

Take any two points A, B in the plane, (§, ŋ, Ŝ) and (x, y, z), and, in the line joining them, consider a point P, (x', y', z′), which divides the distance between them in the ratio μv; so that AP BP v : μ ;

let us suppose P, (x, y, z) to be on the curve (55); then substituting (56) in (55), and suppressing the common denominator (μ+v)", which may be done because (55) is homogeneous, we have F (ν + μξ, vy + μη, vz + μζ) = 0;

(57)

which is an equation of the nth degree in terms of the ratio μ : ν. The roots of this are the ratios in which the line joining (§, n, ¿) and (x, y, z) are cut by the curve at each of the n points where it meets the curve; and if these n values are substituted in (56), we shall have the coordinates to the n points of section. Let (57) be expanded by the theorem given in (56), Art. 142; then, as (57) is homogeneous and of the nth degree, we have

Let us take a more convenient notation; and let

2-1

1

-2

1.2

dy

dz

vn-3μ2 p2+ ... + vμ"-1P1-1 + μ"p1=0; (60)

which is again of the nth degree in terms of the ratio μ: v, and of which the roots refer to the n points of intersection of the straight line with the curve.

The extreme point (x, y, z) of this cutting line is not fixed; it may therefore be any point in the plane of the curve: let us assume it to be on the curve: then it is manifest that p = 0, and also that one of the roots of (60) is equal to zero, or that the equation is divisible by μ.

If the line meets the curve in two coincident points, let us suppose them to be (x, y, z) the extremity of the cutting line; then two values of μ are equal to zero, and (60) must be divisible by μ2. In this case p = 0, and p1 = 0: and as these are simultaneous, the points to which these equal and vanishing values of μ correspond are the points of intersection of the two curves pF (X, Y, Z) = 0, and p1 = 0. But if a line drawn from (§, n, C) to the curve meets the curve in two coincident points, that line is a tangent: therefore the equation p1 = 0, which is of the first order in terms of §, ŋ, %, is that of the tangent of the curve; and this expressed at length is

which is the same equation as that found above in Art. 216, and is of the (n-1)th order in terms of x, y, z; so that the points at which tangents drawn from (§, ŋ, ¿) touch the original curve lie on a curve of the (n-1)th order. And this result admits of extension; for from (59) it appears that p2 = 0, which is an equation of the (n-1)th order in terms of x, y, z, is that which passes through the (n-2) points at which tangents drawn

from (,, ) touch the curve p1 = 0; similarly, p30 is the curve of the (n-3)rd order which passes through the (n−3) points of contact of lines drawn from (§, ~, ¿) with the curve P2 = 0; and similarly P2-10 is the equation to a straight line in terms of x, y, z which passes through the points of contact of the tangents drawn from (§, n,) to the conic Pn-20. The further consideration of (59) and (60) must be deferred to a subsequent Section.

225.] In connexion with the present subject it remains for us still to investigate more generally the equation of the system or the pencil of tangents which can be drawn from a given point (§, n, ¿) to a curve of the nth degree.

,η,

Let the equation to the curve be r (x, y, z) = 0; then we have the following equations,

of which (63) is the equation to the tangent, and (64) is the equation of a straight line passing through (§, 7, 8) and another point (x', y', z′), and of which line therefore x', y', z′ may be the current coordinates. Now (62) is a homogeneous equation of n dimensions, (63) is also homogeneous and of n − 1 dimensions in terms of x, y, z; and (64) is of one dimension in terms of the same variables; therefore, if x, y, z are eliminated, the resultant is, see Art. 209, of n (n-1) dimensions in terms of nz' —¿y', ¿x' — §ź, ¿y' — ŋx'; of n dimensions in terms of ,, ; and apparently of 2n-1 dimensions in terms of the coefficients of r = O, because these coefficients enter in the nth degree by reason of (63) and in the (n-1)th degree by reason of (62); but the resultant has a linear rational factor with reference to the coefficients of r = 0, because (a', y', ) is on the tangent, and

this factor is therefore to be omitted; and the resultant consequently is of the 2 (n-1)th degree with respect to the coefficients of F = 0; and is the equation of the system of tangents which can be drawn to the curve from the point (È, 7, 5). As