# Parabola Sentence Examples

- It was investigated by Galileo, who erroneously determined it to be a
**parabola**; Jungius detected Galileo's error, but the true form was not discovered until 1691, when James Bernoulli published it as a problem in the Aeta Eruditorum. - By joining the points so obtained the
**parabola**may be described. - - The
**parabola**affords a simple example of the use of infinitesimals. - 6) be any arc of a
**parabola**; and suppose we require the area of the figure bounded by this arc and the chord AB. - Thus we find from (i) that Simpson's second formula, for the case where the top is a
**parabola**(with axis, as before, at right angles to the base) and there are three strips of breadth h, may be replaced by area = 8h(3u i + 2U 1 + 3us). - - Geometrical constructions of the
**parabola**are to be found in T. - In the geometry of plane curves, the term
**parabola**is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian**parabola**; a 2 x = y 3 is the cubic**parabola**, a 3 x = y4 is the biquadratic**parabola**; semihave the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical**parabolas****parabola**and ax 3 = y 4 the semibiquadratic**parabola**. - And at equal horizontal intervals, the vertices of the funicular will lie on a
**parabola**whose axis is vertical. - While resembling the
**parabola**in extending to infinity, the curve has closest affinities to the ellipse. - Focus by two tangents drawn from a point), and (having given the focus and a double ordinate) he uses the focus and directrix to obtain any number of points on a
**parabola**- the first instance on record of the practical use of the directrix. - Ann., 1880, 11, 399) that in weak fields the relation of the magnetization I to the magnetizing force H is approximately expressed by an equation of the form I =aH +bH2, or K=I/H =a+bH, whence it appears that within the limits of Baur's experiments the magnetization curve is a
**parabola**, and the susceptibility curve an inclined straight line, x being therefore a known function of H. - Similarly, the inertia parallel to Oy and Oz is NW' - 1 B W', B C (b2 +-X, c 2 ab and A +C abc/ZP, Ao For a sphere a=b=c, Ao= Bo=Co =, 'a' = Q = = z, (9) U from (II), (16) so that the effective inertia of a sphere is increased by half the weight of liquid displaced; and in frictionless air or liquid the sphere, of weight W, will describe a
**parabola**with vertical acceleration W - W', g (30) W+ aW Thus a spherical air bubble, in which W/W' is insensible, will begin to rise in water with acceleration 2g. - It may be regarded as an epicycloid in which the rolling and fixed circles are equal in diameter, as the inverse of a
**parabola**for its focus, or as the caustic produced by the reflection at a spherical surface of rays emanating from a point on the circumference. - Draw the tangents at A and B, meeting at T; draw TV parallel to the axis of the
**parabola**, meeting the arc in C and the chord in V; and M draw the tangent at C, meeting AT and BT in a and b. - Similarly, for a corresponding figure K'L'BA outside the
**parabola**, the area is lK'L'(K'A+4M'C +L'B). - - In § 23 the area of a right trapezium has been expressed in terms of the base and the two sides; and in § 34 the area of a somewhat similar figure, the top having been replaced by an arc of a
**parabola**, has been expressed in terms of its base and of three lengths which may be regarded as the sides of two separate figures of which it is composed. - The top is then a
**parabola**whose axis is at right angles to the base; and the area can therefore (§ 34) be expressed in terms of the two bounding ordinates and the midordinate. - Hence, for the case of a
**parabola**, we can express the area in terms of the bounding ordinates of two strips, but, if we use mid-ordinates, we require three strips; so that, in each case, three ordinates are required. - These problems were also attacked by the Arabian mathematicians; Tobit ben Korra (836-901) is credited with a solution, while Abul Gud solved it by means of a
**parabola**and an equilateral hyperbola. - A solution by means of the
**parabola**and hyperbola was given by Dionysodorus of Amisus (c. 1st century B.c), and a similar problem - to construct a segment equal in volume to a given segment, and in surface to another segment - was solved by the Arabian mathematician and astronomer, Al Kuhi. - As the load travels, the shear at the head of the train will be given by the ordinates of a
**parabola**having its vertex at A, and a maximum F max. - Now, except for very short bridges and very unequal loads, a
**parabola**can be found which includes the curve of maximum moments. - This
**parabola**is the curve of maximum moments for a travelling load uniform per ft. - Experience shows that (a) a
**parabola**having the same ordinate at the centre of the span, or (b) a**parabola**having 15 ons FIG. - 3 parallel to AM; the energycurve AQE would be another straight line through A; the velocitycurve AvV, of which the ordinate v is as the square root of the energy, would be a
**parabola**; and the acceleration of the shot being constant, the time-curve AtT will also be a similar**parabola**. - If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a
**parabola**curving downwards, and the velocity-curve can be represented by an ellipse, or circle with centre F and radius FA; while the time-curve will be a sinusoid. - But if the pressure-curve is a straight line F'CP sloping upwards, cutting AM behind A in F', the energy-curve will be a
**parabola**curving upwards, and the velocity-curve a hyperbola with center at F'. - He extended the "law of continuity" as stated by Johannes Kepler; regarded the denominators of fractions as powers with negative exponents; and deduced from the quadrature of the
**parabola**y=xm, where m is a positive integer, the area of the curves when m is negative or fractional. - It is clearly the form of the fundamental property (expressed in the terminology of the "application of areas") which led him to call the curves for the first time by the names
**parabola**, ellipse, hyperbola. - He went on to deal with the case of projectiles, and was led to the conclusion that the motion in this case could be regarded as the result of superposing a horizontal motion with uniform velocity and a vertical motion with constant acceleration, the latter identical with that of a merely falling body; the inference being that the path of a projectile would be a
**parabola**except for deviations attributed to contact with the air, and that in a vacuum this path would be accurately followed. - In this case the curve representing the distribution of temperature is a
**parabola**, and the conductivity k is deduced from the mean rise of temperature (R-R°)/aR° by observing the increase of resistance R-R° of the bar, and the current C. It is also necessary to measure the cross-section g, the length 1, and the temperature-coefficient a for the range of the experiment. - (9) By a property of the
**parabola**, the mean temperature is 3rds of the maximum temperature, we have therefore (R - R 0)/aRo =1C 2 Ra/12gk, (io) which gives the conductivity directly in terms of the quantities actually observed. **PARABOLA**, a plane curve of the second degree.- The general relations between the
**parabola**, ellipse and hyperbola are treated in the articles Geometry, Analytical, and Conic Sections; and various projective properties are demonstrated in the article Geometry, Projective. - Here only the specific properties of the
**parabola**will be given. - Any number of points on the
**parabola**are obtained by taking any point E on the directrix, joining EG and EF and drawing FP so that the angles PFE and DFE are equal. - It also follows that a line half-way between a point and its polar and parallel to the latter touches the
**parabola**, and therefore the lines joining the middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the ninepoint circle of a self-conjugate triangle passes through the focus. - In the article Geometry, Analytical, it iS Shown that the general equation of the second degree represents a
**parabola**when the highest terms form a perfect square. - More convenient forms in terms of a single parameter are deduced by substituting x' =am t, y' = aam (for on eliminating in between these relations the equation to the
**parabola**is obtained). - Try = o to be a
**parabola**is lbc+mca+nab = o, and the conic for which the triangle of reference is self-conjugate la 2 +143 2 +n7 2 =o is a 2 inn--+b 2 nl+c 2 lm=o. - The cartesian
**parabola**is a cubic curve which is also known as the trident of Newton on account of its three-pronged form. - Parallel and are bisected by the same vertical line; and a
**parabola**with vertical axis can therefore be described through A, B, C, D. - The same holds for the four points B, C, D, E and so on; but since a
**parabola**is uniquely determined by the direction of its axis and by three points on the curve, the successiveABCD, BCDE, CDEF ...**parabolas** - The graph of F is a straight line; that of M is a
**parabola**with vertical axis. - This property is characteristic of a
**parabola**whose axis is vertical. - Which is the equation of the
**parabola**in question. - The curve of positions corresponding to (4) is a
**parabola**, and that of velocities is a straight line.