of this line with respect to (1), we have the centre of the circle touching the three given circles. 318. To find the equation of the reciprocal of a conic with regard to its centre. We found, in Art. 178, that the perpendicular on the tangent could be expressed in terms of the angles it makes with the axes, p2=a* cos20+b2 sin2 0. Hence the polar equation of the reciprocal curve is a concentric conic, whose axes are the reciprocals of the axes of the given conic. 319. To find the equation of the reciprocal of a conic with regard to any point (x'y'). The length of the perpendicular from any point is (Art. 178) therefore, the equation of the reciprocal curve is (xx' +yy' + k2)2 = a2x2 +b2y”. 320. Given the reciprocal of a curve with regard to the origin of co-ordinates, to find the equation of its reciprocal with regard to any point (x'y'). If the perpendicular from the origin on the tangent be P, the perpendicular from any other point is (Art. 34) P-x' cose-y' sine, and, therefore, the polar equation of the locus is xx' + yy' + k2 2 we must, therefore, substitute, in the equation of the given R cos 0 P cos == and The effect of this substitution may be very simply written as follows: Let the equation of the reciprocal with regard to the origin be un where un denotes the terms of the nth degree, &c., then the reciprocal with regard to any point is un + Un-1 (xx' + yy' + k2, +Un-2 'xx'+yy' + k2· 2 + &c. = 0, a curve of the same degree as the given reciprocal. 321. To find the reciprocal with respect to x2+y3 — k2 of the conic given by the general equation. We find the locus of a point whose polar xx' +yy' — k2 shall touch the given conic by writing x', y', - k2 for λ, μ, v in the tangential equation (Art. 151). The reciprocal is therefore Ax2 + 2Hxy + By3 – 2 Gk3x – 2Fk3y + Ck* = 0. == 2 Thus, if the curve be a parabola, C or ab-h2 = 0, and the reciprocal passes through the origin. We can, in like manner, verify by this equation other properties proved already geometrically. If we had, for symmetry, written 2=-z2, and looked for the reciprocal with regard to the curve x2 + y2+z2= 0, the polar would have been xx'+yy'+zz', and the equation of the reciprocal would have been got by writing x, y, z for λ, μ, v in the tangential equation. In like manner, the condition that λx+μy+vz may touch any curve, may be considered as the equation of its reciprocal with regard to x2 + y2 + z2. A tangential equation of the nth degree always represents a curve of the nth class; since if we suppose λx+μy + vz to pass through a fixed point, and therefore have λx' + μy' + vz' = 0; eliminating between this equation and the given tangential equation, we have an equation of the nth degree to determine μ; and therefore n tangents can be drawn through the given point. 322. Before quitting the subject of reciprocal polars, we wish to mention a class of theorems, for the transformation of which M. Chasles has proposed to take as the auxiliary conic a parabola instead of a circle. We proved (Art. 211) that the intercept made on the axis of the parabola between any two lines is equal to the intercept between perpendiculars let fall on the axis from the poles of these lines. This principle, then, enables us readily to transform theorems which relate to the magnitude of lines measured parallel to a fixed line. We shall give one or two specimens of the use of this method, premising that to two tangents parallel to the axis of the auxiliary parabola correspond the two points at infinity on the reciprocal curve, and that, consequently, the curve will be a hyperbola or ellipse, according as these tangents are real or imaginary. The reciprocal will be a parabola if the axis pass through a point at infinity on the original curve. "Any variable tangent to a conic intercepts on two parallel tangents, portions whose rectangle is constant." To the two points of contact of parallel tangents answer the asymptotes of the reciprocal hyperbola, and to the intersections of those parallel tangents with any other tangent answer parallels to the asymptotes through any point; and we obtain, in the first instance, that the asymptotes and parallels to them through any point on the curve intercept on any fixed line portions whose rectangle is constant. But this is plainly equivalent to the theorem: "The rectangle under parallels drawn to the asymptotes from any point on the curve is constant." Chords drawn from two fixed points of a hyperbola to a variable third point, intercept a constant length on the asymptote. (p. 179). If any tangent to a parabola meet two fixed tangents, perpendiculars from its extremities on the tangent at the vertex will intercept a constant length on that line. This method of parabolic polars is plainly very limited in its application. CHAPTER XVI. HARMONIC AND ANHARMONIC PROPERTIES OF CONICS.* 323. THE harmonic and anharmonic properties of conic sections admit of so many applications in the theory of these curves, that we think it not unprofitable to spend a little time in pointing out to the student the number of particular theorems either directly included in the general enunciations of these properties, or which may be inferred from them without much difficulty. AB Вс when The cases which we shall most frequently consider are, when one of the four points of the right line, whose anharmonic ratio we are examining, is at an infinite distance. The anharmonic ratio of four points, A, B, C, D, being in general AB AD (Art. 56) = reduces to the simple ratio BC ᎠᏟ D is at an infinite distance, since then AD ultimately - DC. If the line be cut harmonically, its anharmonic ratio = -1; and if D be at an infinite distance AB=BC, and AC is bisected. The reader is supposed to be acquainted with the geometric investigation of these and the other fundamental theorems connected with anharmonic section. 324. We commence with the theorem (Art. 146): "If any line through a point O meet a conic in the points R', R", and the polar of O in R, the line OR'RR" is cut harmonically." First. Let R" be at an infinite distance; then the line OR must be bisected at R'; that is, if through a fixed point a line be drawn parallel to an asymptote of an hyperbola, or to a diameter of a parabola, the portion of this line between the fixed point and its polar will be bisected by the curve (Art. 211). *The fundamental property of anharmonic pencils was given by Pappus, Math. Coll. VII., 129. The name "anharmonic" was given by Chasles in his History of Geometry, from the notes to which the following pages have been developed. Further details will be found in his Traité de Géométrie Supérieure; and in his recently published Treatise on Conics. The anharmonic relation, however, had been studied by Mobius in his Barycentric Calculus, 1827, under the name of "Doppelschnittsverhältniss." Secondly. Let R be at an infinite distance, and R'R" must be bisected at 0; that is, if through any point a chord be drawn parallel to the polar of that point, it will be bisected at the point. If the polar of O be at infinity, every chord through that point meets the polar at infinity, and is therefore bisected at 0. Hence this point is the centre, or the centre may be considered as a point whose polar is at infinity (p. 150). Thirdly. Let the fixed point itself be at an infinite distance, then all the lines through it will be parallel, and will be bisected on the polar of the fixed point. Hence every diameter of a conic may be considered as the polar of the point at infinity in which its ordinates are supposed to intersect. This also follows from the equation of the polar of a point (Art. 145) y’ (ax+hy+g) + (hx+by+f) 2 + 9x+fy + c x' = 0. Now, if x'y' be a point at infinity on the line mynx, we must make = х becomes and x′ infinite, and the equation of the polar m (ax+hy+g) + n (hx + by +ƒ) = 0, a diameter conjugate to my = nx (Art. 141). 325. Again, it was proved (Art. 146) that the two tangents through any point, any other line through the point, and the line to the pole of this last line, form a harmonic pencil. If now one of the lines through the point be a diameter, the other will be parallel to its conjugate, and since the polar of any point on a diameter is parallel to its conjugate, we learn that the portion between the tangents of any line drawn parallel to the polar of the point is bisected by the diameter through it. Again, let the point be the centre, the two tangents will be the asymptotes. Hence the asymptotes, together with any pair of conjugate diameters, form a harmonic pencil, and the portion of any tangent intercepted between the asymptotes is bisected by the curve (Art. 196). 326. The anharmonic property of the points of a conic (Art. 259) gives rise to a much greater variety of particular theorems. For, the four points on the curve may be any whatever, and |