an example of this process let us take the ellipse whose equation is x2 y2 a2 + + = 0; (66) and the equation to the line through (§, 7, () and (x', y', z′) is x (¿Y1−nz1) + Y (§ Z1 − (x1) +- ≈ (nx1 — Ey1) = 0; (68) 12 a2 62 which is a quadratic equation in terms of a', y', z', and gives therefore two values for the point to which a line drawn from (§, 1, 5) will be a tangent to the ellipse: two tangents therefore can be drawn to an ellipse from a given point. Further investigations in connexion with this subject will be found in a paper by Joachimsthal in Crelle's Journal, Vol. XXXIII, p. 371, and in a paper by Mr. Cayley in the same Journal, Vol. XXXIV, p. 30. 226.] The equation to the tangent given in (18) or (61), which is equivalent also to equation (52), must still be considered in a more general way, and with reference to the curve which passes through the n (n-1) points of contact: for, as before observed, these equations are of the (n-1)th degree in terms of x, y, and z, and of the first degree in reference to §, 7, 5. This curve requires a peculiar name on account of many properties which it possesses; and it is called the Polar, or the first polar, with reference to the point (§, 7, 8) which is called the Pole. The original curve r(x, y, z) = 0 is called the Basecurve; and thus Base-curve, Pole, and Polar are correlative terms. In (59), if (x, y, z) is a point on the curve p = r (x, y, z) = 0, p is the base-curve: Pi = = 0 or is the equation of the first polar, with reference to (§, 7, () as the pole. Thus if the base-curve is a conic of which the equation is Ax2+By2+C≈2+2x y z +26 zx+2н xу = 0, the equation of the first polar is § (AX + HY + GZ) + n (H x +BY+EZ) + Č (GX+EY + cz) = 0, which is of the first degree in terms of x, y, z, and is therefore the equation to a straight line. Similarly if the base-curve is of the third degree, the first polar is a conic, and all the six points of contact of tangents to the curve drawn from a given point are on a conic. Let us consider the modifications which (71) undergoes for particular positions of the pole. Let the origin be the pole; then = n = 0; and (71) becomes and this is the polar of the origin. (72) Thus the polar of the origin of the conic whose equation is (51), Art. 208, is GX+EY + Cz = 0. (73) Again, let us suppose the pole to be at an infinite distance, so that all the tangents drawn from it to the curve are parallel, and let us suppose b a to be the tangent of the angle which all these parallel tangents make with the axis of ; then the equation to the polar is which is of the (n-1)th degree; and as the points of contact are the n (n-1) points, common to this and to the original curve, it follows that all the points of contact of parallel tangents lie on a curve of the (n-1)th degree. If all the tangents are parallel to the axis of x, b = 0, and therefore (dr) = 0; similarly if the tangents are parallel to dx the axis of y, a = 0, and therefore dr dr = 0. Thus = 0, dy dx = 0 are the equations to the polars of r (x, y, z) = 0, and (d): dy when the pole is at an infinite distance on the axes of x and y respectively. One case of these equations requires notice: if r (x, y, z) is the equation of a conic, (d) and dx pressions of the first degree, and represent straight lines. Now if dr = 0, at the points common to it and the conic, the tandx gents of the conic are parallel to the axis of a; and as all chords of a conic parallel to these parallel tangents are bisected by the line passing through the points of contact, so = 0 is the equation of a line bisecting all chords parallel to the axis of x; it is therefore a diameter. Similarly (d) =0 is the equation to a diameter bisecting all chords parallel to the axis of y: accordingly the point of intersection of these two lines is the centre of the conic. See Ex. 2, Art. 141. If F(x, y, z) is of three dimensions, (74) is the equation to a conic; and therefore the six points in a curve of the third degree at which lines parallel to a given straight line touch the curve lie in a conic. 227.] Suppose that the pole is not a fixed point, but moves in a given curve in the plane of the base-curve: then as the position of the pole moves, that also of the polar is changed. Let us call the curve along which the pole moves, the Directrix : then, if the directrix is a continuous curve, the successive corresponding polars will have positions infinitesimally consecutive, and will doubtless in their consecutive intersections generate another curve: curves generated in this way are called Envelopes; but the general theory of them must be reserved to a future part of our treatise. See Chapter XIII. Without entering on the general theory I am here able to consider the most simple case; that, viz. in which the directrix is a straight line. Let the equation to the directrix be a § + bn + c} = 0; the equation to the first polar is (75) η and as no other relation is given between § 7 and 5, these quantities are indeterminate; and thus we have which are two equations of the (n-1)th degree in terms of x, y, z: these are the equations to two curves each of which is of the (n-1)th order; and these curves do not involve έ, 7, 5: they are therefore the same for all positions of the pole in the directrix; and they intersect in (n−1)2 points; and therefore we conclude that all the first polars of a curve of the nth degree pass through (n-1)2 points, if the directrix of the pole is a straight line. A particular case of this is the well-known theorem, that all the polars of a conic corresponding to poles on a given straight line pass through one and the same point. 228.] The first polar is a curve of the (n−1)th degree. To it let a process with reference to the same pole be applied similar to that by which the first polar is derived from the base-curve: then another curve will be derived from it, and this may from analogy be called the second polar, being as it is the polar of the aforesaid first polar, and with reference to the same pole. Similarly may other polars, the third, the fourth, &c. be derived. On referring to the series of equations (59) P1 = 0 is the equation of the first polar; and as p2 = 0 bears the relation to it, with reference to (, n, ), which p1 = 0 does to P1 the base-curve, p2 = 0 is the equation to the second polar: P2 similarly p3 = 0 is the equation to the third polar; and so on; and as p = 0 is an equation of the nth degree in terms of x, y, z, so if curves are formed according to the scheme in (59), Pn-1=0 is of the first degree in terms of x, y, z; and is therefore as to these coordinates the equation of a straight line. Similarly Pn-20 is the equation of a conic; therefore the (n-1) th polar is a straight line, and the (n-2)th is a conic. These results are true, whatever is the base curve. I may also observe that by reason of the symmetry of the function in (57), = which are respectively the equations to the polar conic and the polar straight line. It is also to be observed that the successive polars P1, P2, ... P-1, are in terms of έ, n, severally of the orders 1, 2,... (n − 1) *. 229.] The general equation of the normal to a plane curve given in (21), Art. 217, is which is evidently of n dimensions in terms of x and y, and does not generally admit of reduction. If (§, 7) is a point from which normals are drawn to a curve F(x, y) of the nth degree, (x, y) in (81) is the point at which the normal meets the curve: and as (81) is of n dimensions, it follows that there are as many points of intersection of normals with the curve as there are points of intersection of (81) with the original curve: that is, the number of points is n2: therefore from a given point (§,ŋ) n2 normals may be drawn to a curve: this is clearly the largest number; and the number which can actually be drawn will be the same as that of the real roots in the equation of n2 dimensions which arises from the combination of (81) and of F(x, y) = 0. As έ and ŋ enter into this equation by means of (81), so will the nature of the roots depend on them; and therefore there may be certain districts of the plane of reference, in which if the point (§, n) is taken, all the roots may be real: certain lines of demarcation on which if (§, ŋ) is, two or an even or any number of the roots may become equal; and certain districts again beyond these lines for which pairs of the roots will be imaginary. Thus to a central conic from all points within certain districts four normals can be drawn; from points on a certain line, which is called the Evolute, two of the four roots which give the four normals will be equal and only three normals can be drawn; and from all points in the district beyond this line only two normals can be drawn. To the parabola not more than three normals can be drawn from any point in its plane, the fourth normal being that to the infinite branch of the curve. Thus if * For other theorems as to successive polars, see a memoir by Steiner in Crelle's Journal, Vol. XLVII, p. 1; and the French translation of it in Liouville, Vol. XVIII. |