Elements of the Conic SectionsR. Chapman, 1817 - 253 pàgines |
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abscissa bola BOOK BOOK III centre cone conical surface conjugate diameters conjugate hyperbolas consequently described diame directrix distance drawn parallel Elem equiangular excess foci focus fore given in position greater axis half the second hyper intercepted join latus rectum let it meet line AE line be drawn lipsis meet the hyperbola meeting the asymptotes opposite hyperbolas parallelogram perbola perpendicular plane point D point G point of contact PROP proposition rallel rectangle AKB rectangle contained right angles second axis segments semidiameter square of CB straight line AC straight line drawn straight line parallel straight line touching straight lines ordinately tangent THEOR touches the ellipsis touches the hyperbola touches the parabola transverse axis Aa transverse diameter triangle vertex vertices
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Pàgina 230 - ... upon geometrical loci ; give me leave to present one to your notice, to which I should like to have your demonstration, to compare with my own. ***** When a circle and a right line are given in position, can you determine a point in the circumference of the circle, from which a tangent being drawn ; the segment of the tangent intercepted between the point of contact, and the line shall be given in length ? The enunciation you will excuse, provided you can comprehend me ; we may sometimes dispense...
Pàgina 56 - ... the surface generated is called a conic surface, and the solid terminated by the surface is called a cone. II. The point is called the vertex, and the circle the base of the cone. The straight line drawn from the vertex to the centre of the circle, is called the axis. If the axis be perpendicular to the plane of the base, the cone is said to be right. III. If the generating line be produced indefinitely in both directions, it is evident that the surface will extend indefinitely both above and...
Pàgina 10 - ... string FGC : take away the common part FG, and the remainder EG will be equal to the remainder GC. COROLLARY. »Hence that segment of the axis which is intercepted between the focus and the directrix, is bisected in the vertex of the axis. Thus CB is bisected in H. PROP. II. THEOR. If the distance of any point from the focus of a parabola be equal to the: perpendicular drawn from the same point to the directrix, that point is in the parabola.
Pàgina 247 - Let there be a cone whose vertex is the point A and base the circle BF, and let it be cut by a plane through the axis, and let the section so made be the...
Pàgina 30 - If from a point without a circle there be drawn two straight lines, one of which is perpendicular to a diameter, and the other cuts the circle; the square of the perpendicular is equal to the rectangle contained by the whole cutting line and the part without the circle, together with .the rectangle contained by the segments of the diameter.
Pàgina 106 - The squares of straight lines ordinately applied to the same diameter, are to one another as the rectangles contained by the segments of that diameter, as was demonstrated (1.
Pàgina 59 - It is plain, from the definition of a surface of revolution, that every point of the generatrix will describe the circumference of a circle, the centre of which is in the axis of revolution.
Pàgina 140 - Also a diameter parallel to a straight line ordinately applied to another diameter, it is said to be ordinately applied to this other diameter. IX. A straight line which meets the hyperbola in only one point, and which, being produced both ways, falls without the opposite hyperbolas, is said to touch the hyperbola in that point. PROP. I. THEOR. If from a point in...
Pàgina 194 - S3. If an ordinate be drawn to a second diameter of opposite hyperbolas : the square of this second diameter is to the square of the conjugate diameter, as the sum of the squares of half the second diameter, and the part of it between the ordinate and the centre, is to the square of the ordinate. Let AB and...
Pàgina 78 - To find the solid content of a segment of a spheroid. RULE. Find the spherical segment which has the same height and the same axis ; then, if the base be perpendicular to the fixed axis, the square of that axis is to the square of the other as the spherical to the spheroidal segment. But if the revolving axis be perpendicular to the base, that axis is to the fixed one as the spherical to the spheroidal segment.