or RW: P'V:: CD: CP2. Now, when the circle becomes the circle of curvature at P, the points R and Q move up to, and coincide with P, and the lines RW and PH become equal, while P'V becomes equal to PP', or 2 CP. Hence, PH: 2CP:: CD2: CP2, PROP. XXX. If PU be the diameter of the circle of curvature at the point P of the hyperbola, and PF be drawn at right angles to CD; then PU. PF-2CD2. Since the triangle PHU is similar to the triangle PFC, .. PU: PH:: CP: PF, .. PU.PF= PH. CP, If PI be the chord of the circle of curvature through the focus of the hyperbola; then Let S'P meet CD in E; then since the triangles PIU and PEF are similar, 67. If P be any point on the hyperbola, and CD be conjugate to CP; then SP. S'P CD2. = Draw PII' parallel to the asymptote CE meeting the directrices in I and I', and CB' in U. Let the ordinates NP, MD meet the asymptote in R, and draw PW perpendicular to the directrix; then by similar triangles, PI: PW: CE: CX, :: CA : CX. (Prop. XVII.) PROBLEMS ON THE HYPERBOLA. 1. THE locus of the centre of a circle touching two given circles is a hyperbola. 2. If on the portion of any tangent intercepted between the tangents at the vertices a circle be described, it will pass through the foci. 3. In a rectangular hyperbola the tangents at the vertices will meet the asymptotes in the circumference of the circle described on SS' as diameter. 4. If from a point P in a hyperbola PII' be drawn parallel to the transverse axis meeting the asymptotes in I and I'; then PI. PI' = A C2. 5. If a circle be inscribed in the triangle SPS', the locus of its centre is the tangent at the vertex. 6. If PN be the ordinate of the point P, and NQ a tangent to the circle described on the transverse axis as diameter, and PM be drawn parallel to QC meeting the axis in M, then MN=BC. 7. If PN be the ordinate of a point P, and NQ be drawn parallel to AP to meet CP in Q, then AQ is parallel to the tangent at P. 8. If a hyperbola and an ellipse have the same foci, they cut one another at right angles. 9. If the tangent at P intersect the tangents at the vertices in R, r, and the tangent at P' intersect them in R', r', then AR. Ar AR'. Ar'. = |