rather than by the two coordinates x'y'. We shall, therefore, find it useful, in discussing properties of the ellipse, to make a substitution similar to that employed (Art. 102) in the case of the circle, and shall write The geometric meaning of the angle & is easily explained. If we describe a circle on the axis major as diameter, and produce the ordinate at P to meet the circle at Q, then the angle QCL=4, for CL=CQ cos Q CL, or a'a cos ; and PL= (Art. 163); or, since QL = a sin, we have y' = b sin þ. b == a QL 230. If we draw through P a parallel PN to the radius CQ, then PM: CQ:: PL: QL:: b: a, PN parallel to CQ is, of course, = a. Hence, if from any point of an ellipse A a line D M N B D' = a be inflected to the minor axis, its intercept to the axis major = b. If the ordinate PQ were produced to meet the circle again in the point Q', it could be proved, in like manner, that a parallel through P to the radius CQ' is cut into parts of a constant length. Hence, conversely, if a line MN, of a constant length, move about in the legs of a right angle, and a point P be taken so that MP may be constant, the locus of P is an ellipse, whose axes are equal to MP and NP. (See Ex. 12, p. 47.) On this principle has been constructed an instrument for describing an ellipse by continued motion, called the Elliptic Compasses. CA, CD' are two fixed rulers, MN a third ruler of a constant length, capable of sliding up and down between them, then a pencil fixed at any point of MN will describe an ellipse. If the pencil be fixed at the middle point of MN, it will describe a circle. (O'Brien's Coordinate Geometry, p. 112.) Q Hence we obtain the following construction. Let the ordinate at the given point P, when produced, meet the semicircle on the axis major at Q, join CQ, and erect CQ' perpendicular to it; then the perpendicular let fall on the axis from Q' will pass through P', a point on the conjugate diameter. P' M' C M Hence, too, can easily be found the coordinates of P' given in Art. 172, for since From these values it appears that the areas of the triangles PCM, P'CM' are equal. Ex. 1. To express the lengths of two conjugate semi-diameters in terms of the angle p. Ans. a'2 = a2 cos2 + b2 sin2; b′2 = a2 sin2p + b2 cos2p. Ex. 2. To express the equation of any chord of the ellipse in terms of and ' (see p. 94). Ans. cos § (p + p') + sin § (P + 4') = cos} (p − p'). a Ex. 3. To express similarly the equation of the tangent. Ex. 4. To express the length of the chord joining two points a, ß, But (Ex. 1) the quantity between the parentheses is the semi-diameter conjugate to that to the point (a + ẞ); and (Ex. 2, 3) the tangent at the point (a + ẞ) is parallel to the chord joining the points a, ß; hence, if b' denote the length of the semidiameter parallel to the given chord, D = 26' sin (a - ẞ). Ex. 5. To find the area of the triangle formed by three given points a, ß, y. ΩΣ = : ab {sin (a — ß) + sin (3 − y) + sin (y — a)} = ab (2 sin § (a — B) cos (a - ẞ) – 2 sin § (a — B) cost (a + B − 2y)} Ex. 6. If the bisectors of sides of an inscribed triangle meet in the centre its area is constant. Ex. 7. To find the radius of the circle circumscribing the triangle formed by three given points a, ß, y. If d, e, f be the sides of the triangle formed by the three points, c'c"c" (These where b', b", "" are the semi-diameters parallel to the sides of the triangle. If c', c", c"" be the parallel focal chords, then (see Ex. 5, p. 213) R2: expressions are due to Mr. MacCullagh, Dublin Exam. Papers, 1836, p. 22.) Ex. 8. To find the equation of the circle circumscribing this triangle. 4ab Ans. x2+ y2. 2 (a2 — b2) x a cos (a + B) cos (B + y) cos § (y + a) 2 (b2 — a2) Y sin f (a + ß) sin § (ß + y) sin} (y + a) = § (a2 + b2) − § (a2 — b2) {cos (a + B) + cos (ẞ + y) + cos (y + a)} From this equation the coordinates of the centre of this circle are at once obtained. Ex. 9. The area of the triangle formed by three tangents is, by Art. 39, ab tan (a — ẞ) tan § (3 − y) tan} (y — a). Ex. 10. The area of the triangle formed by three normals is tan (a - ẞ) tan § (6 − y) tan} (y − a) {sin (6 + y) + sin (y + a) + sin (a +ẞ)}2, consequently three normals meet in a point if sin (6+ y) + sin (y + a) + sin (a + ẞ) = 0. [Mr. Burnside.] Ex. 11. To find the locus of the intersection of the focal radius vector FP with the radius of the circle CQ. Let the central coordinates of P be x'y', of O, xy, then we have, from the similar triangles, FON, FPM, Now, since is the angle made with the axis by the radius vector to the point 0, we at once obtain the polar equation of the locus by writing p cos o for x, p sin for y, and we find Hence (Art. 193) the locus is an ellipse, of which C is one focus, and it can easily be proved that F is the other. Ex. 12. The normal at P is produced to meet CQ; the locus of their intersection is a circle concentric with the ellipse. Ex. 14. If from the vertex of an ellipse a radius vector be drawn to any point on the curve, find the locus of the point where a parallel radius through the centre meets the tangent at the point. = or The tangent of the angle made with the axis by the radius vector to the vertex y' -; therefore the equation of the parallel radius through the centre is and the locus of the intersection of this line with the tangent is, obviously, a = 1, the tangent at the other extremity of the axis. The same investigation will apply, if the first radius vector be drawn through any point of the curve, by substituting a' and b' for a and b; the locus will then be the tangent at the diametrically opposite point. Ex. 15. The length of the chord of an ellipse which touches a confocal ellipse, 2h6'2 the squares of whose semiaxes are a2 — h2, b2 — h2, is ab [Mr. Burnside]. The condition that the chord joining two points a, ẞ should touch the confocal conic is a2 - h2 or sin2 (a + ẞ) = cos2 } (a — ẞ), a262 (Ex. 4). cos2 (a + B) + But the length of the chord is 2hb'2 26' sin (aẞ) = ab = By the help of this Example several theorems concerning chords through a focus may be extended to chords touching confocal conics. Hence also is immediately derived a proof of Ex. 13, p. 212, for OR. OR' is to OS. OS' as the squares of the parallel diameters (Art. 149), and it is here proved that the chords OR- OR', OSOS' are to each other in the same ratio. 232. The methods of the preceding Articles do not apply to the hyperbola. For the hyperbola, however, we may substitute x' = a seco, y' = b tan p, This angle may be represented geometrically by drawing a tangent MQ from the foot of the ordinate M to the circle described on the transverse axis, then the angle QCM= 4, since CMCQ sec QCM. M We have also QM= a tan p, but PM=b tan p. Hence, if from the foot of any ordinate of a hyperbola we draw a tangent to the circle described on the transverse axis, this tangent is in a constant ratio to the ordinate. Ex. If any point on the conjugate hyperbola be expressed similarly y′′ = b seco', "= a tan p', prove that the relation connecting the extremities of conjugate diameters is = '. [Mr. Turner.] SIMILAR CONIC SECTIONS. 233. Any two figures are said to be similar and similarly placed if radii vectores drawn to the first from a certain point are in a constant ratio to parallel radii drawn to the second from another point o. If it be possible to find any two such points O and o, we can find an infinity of others; for, take any point C, draw oc parallel to OC, and in the constant ratio op OP' P then from the similar triangles OCP, ocp, cp is parallel to CP and in the given ratio. In like manner, any other radius vector through c can be proved to be proportional to the parallel radius through C. If two central conic sections be similar and similarly placed, all diameters of the one are proportional to the parallel diameters of the other, since the rectangles OP.OQ, op.oq are proportional to the squares of the parallel diameters (Art. 149). 234. To find the condition that two conics, given by the general equations, should be similar and similarly placed. Transforming to the centre of the first as origin, we find (Art. 152) that the square of any semi-diameter of the first is equal to a constant divided by a cos'0+2h cos sin +b sin30, |