A Treatise on Conic Sections and the Application of Algebra to Geometry

T. Stevenson, 1837 - 130 páginas

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Página 39 - Pappus, the locus of a point whose distance from a given point is in a given ratio to its distance from a fixed...
Página 81 - To find the locus of a point, the difference of whose distances from two fixed points is always equal to a given quantity 2 a.
Página 115 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Página 95 - Application to the tangent and to its construction. The rectangle of the parts of a secant, comprised between a point of the curve and the asymptotes, is equal to the square of half of the diameter to which the secant is parallel. Form of the equation of the hyperbola referred to its asymptotes. Of the parabola. Axis of the parabola.
Página 5 - AB; and draw a straight line through the middle point of AB at right angles to AB; then it may be easily shewn that this straight line is the required locus. 49. Required the locus of the vertices of all triangles on a given base AB, such that the square on the side terminated at A may exceed the square on the side terminated at B, by a given square. Suppose C to denote a point on the required locus ; from C draw a perpendicular on the given base, meeting it, produced if necessary, at D. Then the...
Página 11 - LEMMA X. The spaces, described from rest by a body acted on by any finite force, are in the beginning of the motion as the squares of the times, in which they are described.
Página 21 - If a body move in any orbit about a fixed centre of force, the areas, described by lines drawn from the centre to the body, lie in one plane, and are proportional to the times of describing them...
Página 21 - AB ; and these triangles are in the same plane, as no force has acted to draw the body out of the plane SAB. Similarly, if impulses be communicated at the end of every interval of T", in directions tending always to S, causing the body to describe CD, DE, &c. in the third, fourth, &c. intervals, the triangles SAB, SBC, SCD, &c. will be all equal, and will lie in the same plane ; and their bases AB, BC, CD, &c. are described in equal times, therefore the area of any number of these triangles, or the...
Página 45 - If the ordinate of P meets the axis in M, and the tangent and normal at P meet the axis in T and N respectively, then MT is the subtangent...
Página 29 - From the preceding equations to the circle, which assume no other property of a circle than that it is the locus of a point which is always at the same distance from a given fixed point, all the theorems relative to the circle established in geometry may readily be deduced.

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