EXAMPLES. 1. If the ordinate of a parabola is 6, and its absciss 15, what is its parameter? Ans. 2.4. 2. The ordinate of a parabola is 20, and its absciss 36; find its parameter. Ans. 11. 3. The abscisses of a parabola are 18 and 32, and the ordinate of the former is 12; required the ordinate of the latter. Ans. 16. 4. Two ordinates are 18 and 24, and the absciss of the former is 18; find that of the latter. Ans. 32. 5. What is the area of a parabola whose base is 25 and altitude 18? Ans. 300. 6. Find the area of a parabola whose double ordinate is 36 and absciss 45. Ans. 1080. 7. What is the area of a parabola whose base is 26 inches, and absciss 9 feet? Ans. 13 feet. 8. What is the area of a parabola whose base is 16 rods, and altitude 10? Ans. 106.666 rods. 9. What is the length of a parabolic curve, the double ordinate being 18 and its altitude 24? Ans. 58.275. 10. The absciss and ordinate are 10 and 8; what is the length of the curve? Ans. 28.09. 11. What is the area of a parabolic zone, the parallel ordinates being 18 and 30, and the altitude 9? Ans. 220.5. 12. What is the area of a parabolic zone, whose parallel ordinates are 6 and 10, and height 6? Ans. 49. 13. The parallel sides of a parabolic zone are 10 and 15, and the distance between them 15; required its area. Ans. 190. All parabolas are similar figures; and hence two parabolas which have the same parameter and absciss are equal to each other. If a parabola and an ellipse have the same vertex and focus, then the major axis of the ellipse is to the distance of the focus from the other vertex, as the parameter of the parabola to the parameter of the ellipse. If a parabola and ellipse have the same focus and parameter, the distances of the vertexes will be in the same ratio. Parallel rays of light falling on the concave surface of a parabola, or a concave surface generated by the revolution of a parabola about its axis, are all collected in the focus, which is called the burning point. All projectiles moving in a vacuum describe a parabolic curve; and heavy bodies, as iron or leaden balls, moving through the air with a velocity not exceeding 1142 feet in a second, describe a parabolic curve very nearly; and on the properties of this curve is founded the theory of gunnery. See ¶ 73. An hyperbola is a plane figure formed by cutting a section from a cone by a plane parallel to its axis, or to any plane within the cone which passes through the cone's vertex. The curve of the hyperbola is such, that the difference between the distances of any point in it from two given points is always equal to a given right line. If the vertexes of two cones meet each other so that their axes form one continuous straight line, and the plane of the hyperbola cut from one of the cones be continued, it will cut the other cone, and form what is called the opposite hyperbola, equal and similar to the former: and the distance between the vertexes of the two hyperbolas is called the major axis, or transverse diameter. Thus AB is the major axis. If the distance between a certain point within the hyperbola, called the focus, and any point in the curve, be subtracted from the distance of said point in the curve from the focus of the opposite hyperbola, the remainder will always be equal to a given quantity, that is, to the major axis: and the distance of either focus from the centre of the major axis is called the eccentricity. The line passing through the centre perpendicular to the major axis, and having the distance of its extremities from those of this axis equal to the eccentricity, is called the minor axis, or conjugate diameter. An ordinate to the major axis, a double ordinate, and an absciss, are to be understood to mean the same as the corresponding lines in the parabola. Thus AB in the figure is the major axis, GC or GR is the eccentricity, C and R being the foci. The line PF is the minor axis, the distances of P and F from A and B being equal to GR, or GC, or AP. EH is an ordinate, and this line produced to meet the curve on the opposite side is called a double ordinate; and AC or AE is an absciss, and BC or BE is a greater absciss. A third proportional to the major and minor axes is called the parameter of the major axis. Thus a third proportional to AB and PF is the parameter of AB, and is equal to the double ordinate passing through the focus at C or R. Properties of the hyperbola. 1. As the semi-major axis, AG, is to the distance of the focus from the centre, GC, so is the distance of any ordinate from the centre, to half the sum of the distances of the point in the curve, on which the said ordinate falls, from the foci of the two hyperbolas. 2. As the square of the major axis, AB2, is to the square of the minor axis, PF2, so is the product of the two abscisses (that is, BE into AE) to the square of the ordinate, EH. 3. As the transverse axis is to the parameter, so is the rectangle of the distances of any ordinate from the vertexes of the two hyperbolas, to the square of that ordinate. Hence, when the transverse, or major axis, and the parameter and absciss are given, the length of any number of ordinates may be found; and if we divide the absciss into several equal parts, and calculate the length of the ordinates to those parts, and divide double the sum of the ordinates by their number, and multiply the quotient by the absciss, we shall obtain the area nearly. 4. If any line be drawn through the focus of the hyperbola, meeting the curve on opposite sides, the rectangle of the two segments between the focus and the curve will equal the rectangle, or product, of half of that line into half of the pa rameter. 5. In an equilateral hyperbola, (that is, one having the two axes equal,) the square of the semi-major axis is equal to the product obtained by multiplying the distance of the focus from the vertex by the distance of the vertex from the focus of the opposite hyperbola. That is, AG ×AG=AC×AR. 6. In an equilateral hyperbola, the product of the distances of any ordinate from the vertexes of the two hyperbolas, is equal to the square of that ordinate. 7. The areas of two hyperbolas having the same major axis and the same absciss, are to each other as their minor axes. 8. As the square of the minor axis is to the square of the major axis, so is the sum of the squares of the semi-minor axis and an ordinate, to the square of the distance between that ordinate and the centre. The sum of this distance and the semi-minor axis will give the greater absciss; and their difference, the less. 9. The product of the abscisses is to the square of the ordinate, as the square of the major axis to that of the minor axis. 10. The minor axis, an ordinate, and two abscisses being given, to find the major axis : Find the square root of the sum of the squares of the semiminor axis and the ordinate; then, according as the less or greater absciss is given, find the sum or difference of this root and the semi-minor axis; then say :-As the square of the ordinate is to the product of the absciss and minor axis, so is the sum or difference found above, to the major axis. 11. To find the length of an arc of an hyperbola, reckoning from the vertex of the curve: To 15 times the major axis add 21 times the less absciss, and multiply the sum by the square root of the minor axis; add this product to 19 times the product of the square of the major axis by the absciss, and add it also to 9 times the same product; divide the former sum by the latter, and multiply the quotient by the ordinate, and the product will be the length of the arc very nearly. 12. To find the area of an hyperbola, the axes and an ordinate being given : To 7 times the major axis add 5 times the less absciss; multiply the sum by 7 times the absciss, and the square root of this product by 3; to this product add 4 times the square root of the product of the major axis and absciss; multiply this sum by 16 times the product of the minor axis and absciss; divide this product by 300 times the major axis, and the quotient will be the required area nearly. EXAMPLES. 1. If the semi-major axis of an hyperbola be 10, and the distance of the focus from the centre of the major axis be 12, and the distance of an ordinate from the centre be 15, what is the sum of the distances of that point in the curve on which the ordinate falls, from the two foci? Ans. 36. |