Elements of analytical geometry and the differential and integral calculus . t value of r is (A-\-c), and the least value (A—c), ob-vious results when the polar point is at F. The above equation may be simplified a little by introducingthe eccentricity. The eccentricity of an ellipse is the distancefrom the center to either focus, when the semi-major axis istaken as unity. Designate the eccentricity by e, then1 : e=A : c. Whence c=eA. Substituting this value ofc in the preceding equation, we have A— 1—e equation is much used in astronomy. CHAPTER Parabola. Definition.—
Elements of analytical geometry and the differential and integral calculus . t value of r is (A-\-c), and the least value (A—c), ob-vious results when the polar point is at F. The above equation may be simplified a little by introducingthe eccentricity. The eccentricity of an ellipse is the distancefrom the center to either focus, when the semi-major axis istaken as unity. Designate the eccentricity by e, then1 : e=A : c. Whence c=eA. Substituting this value ofc in the preceding equation, we have A— 1—e equation is much used in astronomy. CHAPTER Parabola. Definition.—1. A parabola is a plane curve, every point ofwhich is equally distant from a fixed point and a given straightline. 2. The given point is called the focm, and the given line iscalled the directrix. To describe a parabola. Let CD be the given line, and F a gi-ven point. Take a square, as DBG, andto one side of it, GjB, attach a thread,and let the thread be of the same lengthas the side GB of the square. Fasten oneend of the thread at the point G, theother end at THE PARABOLA. 65 Put the other side of the square against the given line, CDyand with a pencil, P, in the thread, bring the thread up to theside of the square. Slide one side of the square along the lineCDy and at the same time keep the thread close against theother side, permitting the thread to slide round the pencil the side of the square, BDy is moved along the line CD, thepencil will describe the curve represented as passing throughthe points Fand P, GP+PF= the +PB=z the thread. By subtraction PF—PB=0, or PF=PJB. This result is true at any and every position of the point P ;that is, it is true for every point on the curve corresponding todefinition 1. Hence, FV=VH. If the square be turned over and moved in the opposite di-rection, the other part of the parabola, the other side of the lineFIImsLy be described. 3. A diameter to a parabola is a straight line drawn throughany point of the cur
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