in the numerator, and OP3 in the denominator, Dividing, then, by these, there will remain merely a relation between the sines of angles subtended at O. It is evident that the points A, B, C, D, E, F, need not be on the same right line; or, in other words, that the perpendicular OP need not be the same for all, provided the points be so taken that after the substitution, each term of the equation may contain in the denominator the same product, OP.OP'.OP", &c. Thus, for example, "If lines meeting in a point and drawn through the vertices of a triangle ABC meet the opposite sides in the points a, b, c, then Ab.Bc. Ca= Ac. Ba. Cb." This is a relation of the class just mentioned, and which it is sufficient to prove for any projection of the triangle ABC. Let us suppose the point C projected to an infinite distance, then AC, BC, Cc are parallel, and the relation becomes the truth of which is at once perceived on making the figure. 352. It appears from what has been said, that if we wish to demonstrate any projective property of any figure, it is sufficient to demonstrate it for the simplest figure into which the given figure can be projected; e.g. for one in which any line of the given figure is at an infinite distance. Thus, if it were required to investigate the harmonic properties of a complete quadrilateral ABCD, whose opposite sides intersect in E, F, and the intersection of whose diagonals is G, we may join all the points of this figure to any point in space 0, and cut the joining lines by any plane parallel to OEF, then EF is projected to infinity, and we have a new quadrilateral, whose sides ab, cd intersect in e at infinity, that is, are parallel; while ad, bc intersect in a point ƒ at infinity, or are also parallel. We thus see that any quadrilateral may be projected into a parallelogram. Now since the diagonals of a parallelogram bisect each other, the diagonal ac is cut harmonically in the points a, g, c, and the point where it meets the line at infinity ef. Hence AB is cut harmonically in the points A, G, C, and where it meets EF. Ex. If two triangles ABC, A'B'C', be such that the points of intersection of AB, A'B'; BC, B'C'; CA, C'A'; lie in a right line, then the lines AA', BB', CC' meet in a point. RR Project to infinity the line in which AB, A'B', &c., intersect; then the theorem becomes: "If two triangles abc, a'b'c' have the sides of the one respectively parallel to the sides of the other, then the lines aa', bb', cc' meet in a point." But the truth of this latter theorem is evident, since aa', bb' both cut cc' in the same ratio. 353. In order not to interrupt the account of the applications of the method of projection, we place in a separate section the formal proof that every curve of the second degree may be projected so as to become a circle. It will also be proved that by choosing properly the vertex and plane of projection, we can, as in Art. 352, cause any given line EF on the figure to be projected to infinity, at the same time that the projected curve becomes a circle. This being for the present taken for granted, these consequences follow: Given any conic section and a point in its plane, we can project it into a circle, of which the projection of that point is the centre, for we have only to project it so that the projection of the polar of the given point may pass to infinity (Art. 154). Any two conic sections may be projected so as both to become circles, for we have only to project one of them into a circle, and so that any of its chords of intersection with the other shall pass to infinity, and then, by Art. 257, the projection of the second conic passing through the same points at infinity as the circle must be a circle also. Any two conics which have double contact with each other may be projected into concentric circles. For we have only to project one of them into a circle, so that its chord of contact with the other may pass to infinity (Art. 257). 354. We shall now give some examples of the method of deriving properties of conics from those of the circle, or from other more particular properties of conics. Ex. 1. "A line through any point is cut harmonically by the curve and the polar of that point." This property and its reciprocal are projective properties (Art. 351), and both being true for the circle, are true for every conic. Hence all the properties of the circle depending on the theory of poles and polars are true for all the conic -sections, Ex. 2. The anharmonic properties of the points and tangents of a conic are projective properties, which, when proved for the circle, as in Art. 312, are proved for all conics. Hence, every property of the circle which results from either of its anharmonic properties is true also for all the conic sections. Ex. 3. Carnot's theorem (Art. 313), that if a conic meet the sides of a triangle, is a projective property which need only be proved in the case of the circle, in which case it is evidently true, since Ab. Ab' = Ac. Ac', &c. The theorem can evidently be proved in like manner for any polygon. Ex. 4. From Carnot's theorem, thus proved, could be deduced the properties of Art. 148, by supposing the point C at an infinite distance; we then have For the line at infinity in the first case is projected into the chord of contact of two conics having double contact with each other. Ex. 4, p. 213, is only a particular case of this theorem. Ex. 6. Given three concentric circles, any tangent to one is cut by the other two in four points whose anharmonic ratio is constant. Given three conics all touching each other in the same two points, any tangent to one is cut by the other two in four points whose anharmonic ratio is constant. The first theorem is obviously true, since the four lengths are constant. The second may be considered as an extension of the anharmonic property of the tangents of a conic. In like manner, the theorem (in Art. 276) with regard to anharmonic ratios in conics having double contact is immediately proved by projecting the conics into concentric circles. Ex. 7. We mentioned already, that it was sufficient to prove Pascal's theorem for the case of a circle, but, by the help of Art. 353, we may still further simplify our figure, for we may suppose the line joining the intersection of AB, DE, to that . of BC, EF, to pass off to infinity; and it is only necessary to prove that, if a hexagon be inscribed in a circle having the side AB parallel to DE, and BC to EF, then CD will be parallel to AF; but the truth of this can be shown from elementary considerations. Ex. 8. A triangle is inscribed in any conic, two of whose sides pass through fixed points, to find the envelope of the third (p. 239). Let the line joining the fixed points be projected to infinity, and at the same time the conic into a circle, and this property becomes,—“A triangle is inscribed in a circle, two of whose sides are parallel to fixed lines, to find the envelope of the third." But this envelope is a concentric circle, since the vertical angle of the triangle given; hence, in the general case, the envelope is a conic touching the given conic in two points on the line joining the two given points. Ex. 9. To investigate the projective properties of a quadrilateral inscribed in a conic. Let the conic be projected into a circle, and the quadrilateral into a parallelogram (Art. 352). Now the intersection of the diagonals of a parallelogram inscribed in a circle is the centre of the circle; hence the intersection of the diagonals of a quadrilateral inscribed in a conic is the pole of the line joining the intersections of the opposite sides. Again, if tangents to the circle be drawn at the vertices of this parallelogram, the diagonals of the quadrilateral so formed will also pass through the centre, bisecting the angles between the first diagonals; hence, "the diagonals of the inscribed and corresponding circumscribing quadrilateral pass through a point, and form a harmonic pencil." Ex. 10. Given four points on a conic, the locus of its centre is a conic through the middle points of the sides of the given quadrilateral. (Ex. 15, p. 290). Ex. 11. The locus of the point where parallel chords of a circle are cut in a given ratio is an ellipse having double contact with the circle. (Art. 163). Given four points on a conic, the locus of the pole of any fixed line is a conic passing through the fourth harmonic to the point in which this line meets each side of the given quadrilateral. If through a fixed point 0 a line be drawn meeting the conic in A, B, and on it a point P be taken, such that {OABP} may be constant, the locus of P is a conic having double contact with the given conic. 355. We may project several properties relating to foci by the help of the definition of a focus given p. 228, viz. that if F be a focus, and A, B the two imaginary points in which any circle is met by the line at infinity; then FA, FB are tangents to the conic. Ex. 1. The locus of the centre of a circle touching two given circles is a hyperbola, having the centres of the given circles for foci. If a conic be described through two fixed points A, B, and touching two given conics which also pass through those points, the locus of the pole of AB is a conic touching the four lines CA, CB, C'A, C'B, where C, C', are the poles of AB with regard to the two given conics. In this example we substitute for the word 'circle,' "conic through two fixed points A, B," (Art. 257), and for the word 'centre," "pole of the line AB." (Art. 154). Ex. 2. Given the focus and two points of a conic section, the intersection of tangents at those points will be on a fixed line. (Art. 191). Ex. 3. Given a focus and two tangents to a conic, the locus of the other focus is a right line. (This follows from Art. 189). Ex. 4. If a triangle circumscribe a parabola, the circle circumscribing the triangle passes through the focus, p. 196. Given two tangents, and two points on a conic, the locus of the intersection of tangents at those points is a right line. Given two fixed points A, B; two tangents FA, FB passing one through each point, and two other tangents to a conic; the locus of the intersection of the other tangents from A, B, is a right line. If two triangles circumscribe a conic, their six vertices lie on the same conic. For if the focus be F, and the two circular points at infinity A, B, the triangle FAB is a second triangle whose three sides touch the parabola. Ex. 5. The locus of the centre of a circle passing through a fixed point, and touching a fixed line, is a parabola of which the fixed point is the focus. Given one tangent, and three points on a conic, the locus of the intersection of tangents at any two of these points is a conic inscribed in the triangle formed by those points. Ex. 6. Given four tangents to a conic, the locus of the centre is the line joining .the middle points of the diagonals of the quadrilateral. Given four tangents to a conic, the locus of the pole of any line is the line joining the fourth harmonics of the points where the given line meets the diagonals of the quadrilateral. It follows from our definition of a focus, that if two conics have the same focus, this point will be an intersection of common tangents to them, and will possess the properties mentioned at the end of Art. 264. Also, that if two conics have the same focus and directrix, they may be considered as two conics having double contact with each other, and may be projected into concentric circles. 356. Since angles which are constant in any figure will in general not be constant in the projection of that figure, we proceed to show what property of a projected figure may be inferred when any property relating to the magnitude of angles is given; and we commence with the case of the right angle. Let the equations of two lines at right angles to each other be x = 0, y = 0, then the equation which determines the direction of the points at infinity on any circle is x2 + y2=0, or x+y√1=0, x-y√-1=0. Hence (Art. 57) these four lines form a harmonic pencil. Hence, given four points, A, B, C, D, of a line cut harmonically, where A, B may be real or imaginary, if these points be transferred by a real or imaginary projection, so that A, B may become the two imaginary points at infinity on any circle, then any lines through C, D will be projected into lines at right angles to each other. Conversely, any two lines at right angles to each other will be projected into lines which cut harmonically the line joining the two fixed points which are the projections of the imaginary points at infinity on a circle. Ex. 1. The tangent to a circle is at right angles to the radius. Any chord of a conic is cut harmonically by any tangent, and by the line joining the point of contact of that tangent to the pole of the given chord. (Art. 146). For the chord of the conic is supposed to be the projection of the line at infinity in the plane of the circle; the points where the chord meets the conic will be the projections of the imaginary points at infinity on the circle; and the pole of the chord will be the projection of the centre of the circle. Ex. 2. Any right line drawn through the focus of a conic is at right angles to the line joining its pole to the focus. (Art. 192). Any right line through a point, the line joining its pole to that point, and the two tangents from the point, form a harmonic pencil. (Art. 146). It is evident that the first of these properties is only a particular case of the |