necting the three perpendiculars λ, μ, v, the trilinear equation of the reciprocal curve is B3 γ where a', ', ' are the trilinear coordinates of the origin. Ex. 1. Given the focus and a triangle circumscribing a conic, the perpendiculars let fall from its vertices on any tangent to the conic are connected by the relation where 0, 0, 0" are the angles the sides of the triangle subtend at the focus. This is obtained by forming the reciprocal of the trilinear equation of the circle circumscribing a triangle. If the centre of the inscribed circle be taken as focus, we have 0 = 90° + §4, p sin A = r, whence the tangential equation, on this system, of the inscribed circle is μv cot 4+ vλ cot B + λu cot C = 0. In the case of any of the exscribed circles two of the cotangents are replaced by tangents. Ex. 2. Given the focus and a triangle inscribed in a conic, the perpendiculars let fall from its vertices on any tangent are connected by the relation The tangential equation of the circumscribing circle takes the form sin A √() + sin B √(u) + sin C √(v) = 0. Ex. 3. Given focus and three tangents the trilinear equation of the conic is This is obtained by reciprocating the equation of the circumscribing circle last found. Ex. 4. In like manner, from Ex. 1, we find that given focus and three points the trilinear equation is tan 10 + tan 10 B + tan jo” 2 = 0. 312. Very many theorems concerning magnitude may be reduced to theorems concerning lines cut harmonically or anharmonically, and are transformed by the following principle: To any four points on a right line correspond four lines passing through a point, and the anharmonic ratio of this pencil is the same as that of the four points. This is evident, since each leg of the pencil drawn from the origin to the given points is perpendicular to one of the corresponding lines. We may thus derive the anharmonic properties of conics in general from those of the circle. The anharmonic ratio of the pencil joining four points on a conic to a variable fifth is constant. The anharmonic ratio of the point in which four fixed tangents to a conic cut any fifth variable tangent is constant. The first of these theorems is true for the circle, since all the angles of the pencil are constant, therefore the second is true for all conics. The second theorem is true for the circle, since the angles which the four points subtend at the centre are constant, therefore the first theorem is true for all conics. By observing the angles which correspond in the reciprocal figure to the angles which are constant in the case of the circle, the student will perceive that the angles which the four points of the variable tangent subtend at either focus are constant, and that the angles are constant which are subtended at the focus by the four points in which any inscribed pencil meets the directrix. 313. The anharmonic ratio of a line is not the only relation concerning the magnitude of lines which can be expressed in terms of the angles subtended by the lines at a fixed point. For, if there be any relation which, by substituting (as in Art. 56) OA. OB.sin A OB for each line AB involved in it, OP can be re duced to a relation between the sines of angles subtended at a given point 0, this relation will be equally true for any transversal cutting the lines joining O to the points A, B, &c.; and by taking the given point for origin a reciprocal theorem can be easily obtained. For example, the following theorem, due to Carnot, is an immediate consequence of Art. 148: "If any conic meet the side AB of any triangle in the points c, c'; BC in a, a'; AC in b, b; then the ratio Now, it will be seen that this ratio is such that we may substitute for each line Ac the sine of the angle 40c, which it subtends at any fixed point; and if we take the reciprocal of this theorem, we obtain the theorem given already Art. 295. 314. Having shown how to form the reciprocals of particular theorems, we shall add some general considerations respecting reciprocal conica. We proved (Art. 308) that the reciprocal of a circle is an ellipse, hyperbola, or parabola, according as the origin is within, PP. without, or on the curve; we shall now extend this conclusion to all the conic sections. It is evident that, the nearer any line or point is to the origin, the farther the corresponding point or line will be; that if any line passes through the origin, the corresponding point must be at an infinite distance; and that the line corresponding to the origin itself must be altogether at an infinite distance. To two tangents, therefore, through the origin on one figure, will correspond two points at an infinite distance on the other; hence, if two real tangents can be drawn from the origin, the reciprocal curve will have two real points at infinity, that is, it will be a hyperbola; if the tangents drawn from the origin be imaginary, the reciprocal curve will be an ellipse; if the origin be on the curve, the tangents from it coincide, therefore the points at infinity on the reciprocal curve coincide, that is, the reciprocal curve will be a parabola. Since the line at infinity corresponds to the origin, we see that, if the origin be a point on one curve, the line at infinity will be a tangent to the reciprocal curve; and we are again led to the theorem (Art. 254) that every parabola has one tangent situated at an infinite distance. 315. To the points of contact of two tangents through the origin must correspond the tangents at the two points at infinity on the reciprocal curve, that is to say, the asymptotes of the reciprocal curve. The eccentricity of the reciprocal hyperbola depending solely on the angle between its asymptotes, depends therefore on the angle between the tangents drawn from the origin to the original curve. Again, the intersection of the asymptotes of the reciprocal curve (ie. its centre) corresponds to the chord of contact of tangents from the origin to the original curve. We met with a particular case of this theorem when we proved that to the centre of a circle corresponds the directrix of the reciprocal conic, for the directrix is the polar of the origin which is the focus of that conic. Ex. 1. The reciprocal of a parabola with regard to a point on the directrix is an equilateral hyperbola. (See Art. 221). Ex. 2. Prove that the following theorems are reciprocal : a triangle circumscribing a parabola is a The intersection of perpendiculars of a triangle inscribed in an equilateral hyperbola lies on the curve. Ex. 3. Derive the last from Pascal's theorem. (See Ex. 3, p. 247). Ex. 4. The axes of the reciprocal curve are parallel to the tangent and normal of a conic drawn through the origin confocal with the given one. For the axes of the reciprocal curve must be parallel to the internal and external bisectors of the angle between the tangents drawn from the origin to the given curve. The theorem stated follows by Art. 189. 316. Given two circles, we can find an origin such that the reciprocals of both shall be confocal conics. For, since the reciprocals of all circles must have one focus (the origin) common; in order that the other focus should be common, it is only necessary that the two reciprocal curves should have the same centre, that is, that the polar of the origin with regard to both circles should be the same, or that the origin should be one of the two points determined in Art. 111. Hence, given a system of circles, as in Art. 109, their reciprocals with regard to one of these limiting points will be a system of confocal conics. The reciprocals of any two conics will, in like manner, be concentric if taken with regard to any of the three points (Art. 282) whose polars with regard to the curves are the same. Confocal conics cut at right angles (Art. 188). The tangents from any point to two confocal conics are equally inclined to each other. (Art. 189). The locus of the pole of a fixed line with regard to a series of confocal conics is a line perpendicular to the fixed line. (Art. 226, Ex. 3). The common tangent to two circles subtends a right angle at either limiting point. If any line intersect two circles, its two intercepts between the circles subtend equal angles at either limiting point. The polar of a fixed point, with regard to a series of circles having the same radical axis, passes through a fixed point; and the two points subtend a right angle at either limiting point. 317. We may mention here that the method of reciprocal polars affords a simple solution of the problem, "to describe a circle touching three given circles." The locus of the centre of a circle touching two of the given circles (1), (2), is evidently a hyperbola, of which the centres of the given circles are the foci, since the problem is at once reduced to-"Given base and difference of sides of a triangle." Hence (Art. 308) the polar of the centre with regard to either of the given circles (1) will always touch a circle which can be easily constructed. In like manner, the polar of the centre of any circle touching (1) and (3) must also touch a given circle. Therefore, if we draw a common tangent to the two circles thus determined, and take the pole 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, p=a" cos'e+b2 sin*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) だ p= √(a* cos*0+ b* sin*0) - x' cos 0-y' sin 0; =- P therefore the equation of the reciprocal curve is (xx' + yy' + k3)* = a*x2 + b*y3. 320. Given the reciprocal of a curve with regard to the origin of coordinates, 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) we must therefore substitute, in the equation of the given k2 reciprocal, k*x for xx' + yy + k2 x, and k'y for y. |