The last theorem may, by Art. 149, be stated otherwise thus: "The locus of a point O, such that the line joining O to the pole of AO may pass through C, is a conic through A, C;" and the truth of it is evident directly, by taking four positions of the line, when we see, by Art. 339, that the anharmonic ratio of four lines, AO, is equal to that of four corresponding lines, CO. Ex. 6. The locus of the intersection of tangents to a parabola, which cut at right angles, is the directrix. Ex. 7. If from any point on a conic two lines at right angles to each other be drawn, the chord joining their extremities passes through a fixed point. (p. 160.) If in the last example AC touch the given conic, the locus of O will be the line joining the points of contact of tangents from A, C. If a harmonic pencil be drawn through any point on a conic, two legs of which are fixed, the chord joining the extremities of the other legs will pass through a fixed point. In other words, given two points, a, e, on a conic, and {abcd} an harmonic ratio, bd will pass through a fixed point, namely, the intersection of tangents at a, c. But the truth of this may be seen directly: for let the line ac meet bd in K, then since {a.abcd} is a harmonic pencil, the tangent at a cuts bd in the fourth harmonic to K: but so likewise must the tangent at c, therefore these tangents meet bd in the same point. As a particular case of this theorem we have the following: "Through a fixed point on a conic two lines are drawn, making equal angles with a fixed line, the chord joining their extremities will pass through a fixed point." 376. A system of pairs of right lines drawn through a point, every two of which make equal angles with a fixed line, cut the line at infinity in a system of points in involution, of which the two points at infinity on any circle form one pair of conjugate points. For they evidently cut any right line in a system of points in involution, the foci of which are the points where the line is met by the given internal and external bisector of every pair of right lines. The two points at infinity just mentioned belong to the system, since they also are cut harmonically by these bisectors. The tangents from any point to a system of confocal conics make equal angles with two fixed lines. (Art. 194.) The tangents from any point to a system of conics inscribed in the same quadrilateral cut any diagonal of that quadrilateral in a system of points in involution of which the two extremities of that diagonal are a pair of conjugate points. (Art. 336.) 377. Two lines diverging from a fixed point, which contain a constant angle, cut the line joining the two points at infinity on a circle, so that the anharmonic ratio of the four points is constant. For the equation of two lines containing an angle 0 being X = 0, y = 0, the direction of the points at infinity on any circle is determined by the equation x2+ y2+ 2xy cos 0 = 0; and, separating this equation into factors, we see, by Art. 55, that the anharmonic ratio of the four lines is constant if 0 be constant. Ex. 1. "The angle contained in the same segment of a circle is constant." We see, by the present Article, that this is the form assumed by the anharmonic property of four points on a circle when two of them are at an infinite distance. Ex. 2. The envelope of a chord of a conic which subtends a constant angle at the focus is another conic having the same focus and the same directrix. Ex. 3. The locus of the intersection of tangents to a parabola which cut at a given angle is a hyperbola having the same focus and the same directrix. Ex. 4. If from the focus of a conic a line be drawn making a given angle with any tangent, the locus of the point where it meets it is a circle. If tangents through any point O meet the conic in T, T', and there be taken on the conic two points A, B, such that {O.ATBT'} is constant, the envelope of AB is a conic touching the given conic in the points T, T. If in Art. 375, Ex. 6, the points B, D be so taken that {ABCD} is constant, the locus of O is a conic touching the given conic at the points of contact of tangents from A, C. If a variable tangent to a conic meet two fixed tangents in T, T, and a fixed line in M, and there be taken on it a point P, such that {PTMT'} may be constant, the locus of P is a conic passing through the points where the fixed tangents meet the fixed line. A particular case of this theorem is: "The locus of the point where the intercept of a variable tangent between two fixed tangents is cut in a given ratio, is a hyperbola whose asymptotes are parallel to the fixed tangents." Ex. 5. If from a fixed point O, OP be drawn to a given circle, and the angle TPO be constant, the envelope of TP is a conic having O for its focus. Given the anharmonic ratio of a pencil three of whose legs pass through fixed points, and whose vertex moves along a given conic, passing through two of the points; the envelope of the fourth leg is a conic touching the lines joining these two to the third fixed point. A particular case of this is: "If two fixed points A, B, on a conic be joined to a variable point P, and the intercept made by the joining chords on a fixed line be cut in a given ratio at M, the envelope of PM is a conic touching parallels through A and B to the fixed line." Ex. 6. If from a fixed point O, OP be drawn to a given right line, and the angle TPO be constant, the envelope of TP is a parabola having O for its focus. Given the anharmonic ratio of a pencil, three of whose legs pass through fixed points, and whose vertex moves along a fixed line, the envelope of the fourth leg is a conic touching the three sides of the triangle formed by the given points.* The method of projections can equally be used in obtaining from properties of plane curves properties of other curves not plane, e. g. curves on the surface of a sphere. Mr. 378. We shall conclude this chapter with a brief account of the method of orthogonal projection, which, before the publication of M, Poncelet's treatise, was the only method of projection much used by geometers. If from all the points of any figure perpendiculars be let fall on any plane, their feet will trace out a figure which is called the orthogonal projection of the given figure. The orthogonal projection of any figure is, therefore, a right section of a cylinder passing through the given figure. All parallel lines are in a constant ratio to their orthogonal projections on any plane. For (see fig. p. 4) MM' represents the orthogonal projection of the line PQ, and it is evidently = PQ multiplied by the cosine of the angle which PQ makes with MM'. All lines parallel to the intersection of the plane of the figure with the plane on which it is projected, are equal to their orthogonal projections. For, since the intersection of the planes is itself not altered by projection, neither can any line parallel to it. The area of any figure in a given plane is in a constant ratio to its orthogonal projection on another given plane. For, if we suppose ordinates of the figure and of its projection to be drawn perpendicular to the intersection of the planes, since every ordinate of the projection is to the corresponding ordinate of the original figure in the constant ratio of the cosine of the Mulcahy, some years ago, gave the following method of obtaining the properties of angles subtended at the focus from those of small circles on a sphere. The method depends on the following principle: the locus of the vertices of all the right cones from which a given ellipse can be cut is a hyperbola passing through the foci of the ellipse. For, see note, p. 305, the difference of MO and NO is constant, being equal to the difference of MF' and NF'. Now, let us take any property of a small circle of a sphere, e. g. if through any point P, on the surface of a sphere, a great circle be drawn, cutting the small circle in the points A, B, then tan AP tan BP is constant. Now, let us take a cone whose base is the small circle, and whose vertex is the centre of the sphere, and let us cut this cone by any plane, and we learn that "if through a point p, in the plane of any conic, a line be drawn cutting the conic in the points a, b, then the product of the tangents of the halves of the angles which ap, bp subtend at the vertex of the cone will be constant; this property will be true of the vertex of any right cone, out of which the section can be cut, and, therefore, since the focus is a point in the locus of such vertices, it must be true that tan afp tan bfp is constant (see p. 191). angle between the planes to unity; by Art. 351, Cor., the areas of the figures will be in the same ratio. Any ellipse can be orthogonally projected into a circle. For, if we take the intersection of the plane of projection with the plane of the given ellipse parallel to the axis minor of that ellipse, and if we take the cosine of the angle between the planes = b a' then every line parallel to the axis minor will be unaltered by projection, but every line parallel to the axis major will be shortened in the ratio b: a, the projection will, therefore (Art. 166), be a circle, whose radius is b. 379. We shall apply the principles laid down in the last Article to investigate the expression for the radius of a circle circumscribing a triangle inscribed in a conic, given Ex. 6, p. 199.* Let the sides of the triangle be a, ß, y, and its area A, then, by elementary geometry, - αβγ R= 4A Now let the ellipse be projected into a circle whose radius is b, then, since this is the circle circumscribing the projected triangle, we have b α' β' γ' = 4A' But, since parallel lines are in a constant ratio to their projections, we have a': ab: b', B': B::b:b", y: y::b:b"; and, since (Art. 378) A' is to A as the area of the circle (= b2) to the area of the ellipse (= Tab), we have * This proof of Mr. Mac Cullagh's theorem is due to Dr. Graves. NOTES. PASCAL'S THEOREM, Page 222. M. STEINER was the first who (in Gergonne's Annales) directed the attention of geometers to the complete figure obtained by joining in every possible way six points on a conic. M. Steiner's theorems were corrected and extended by M. Plücker (Crelle's Journal, vol. v. p. 274), and the subject has been more recently investigated by Messrs. Cayley and Kirkman, the latter of whom, in particular, has added several new theorems to those already known. We shall in this note give a slight sketch of the more important of these, and of the methods of obtaining them. The greater part are derived by joining the simplest principles of the theory of combinations with the following elementary theorems and their reciprocals: "If two triangles be such that the lines joining corresponding vertices meet in a point (which we shall call the pole of the two triangles), the intersections of corresponding sides will lie in one right line (which we shall call their axis)." "If the intersections of opposite sides of three triangles be for each pair the same three points in a right line, the poles of the first and second, second and third, third and first, will lie in a right line.” Now let the six points on a conic be a, b, c, d, e, f, which we shall call the points P. These may be connected by fifteen right lines, ab, ac, &c., which we shall call the lines C. Each of the lines C (for example ab) is intersected by the fourteen others; by four of them in the point a, by four in the point b, and consequently by six in points distinct from the points P (for example the points ab, cd; &c.) These we shall call the points p. There are forty-five such points; for there are six on each of the lines C. To find then the number of points p, we must multiply the number of lines C by 6, and divide by 2, since two lines C pass through every point p. |