Elements of analytical geometry and the differential and integral calculus . gned to «, h, and v,last equation by the equation r=0, and we have <5d Dividing the ^^sin.^i ^^ =0. -| The value of r in this equation is the value of a the chord becomes 0, the value of r in the last equationbecomes 0 also, and then A^ h ;+-B* a Or ——- ,A-b a result corresponding to Prop. VI, as it ought to do, becausethe raditis vector then becomes tangent to the curve. Scholium 3. When P is placed at the extremity of the majoraxis on the right, then ,
Elements of analytical geometry and the differential and integral calculus . gned to «, h, and v,last equation by the equation r=0, and we have <5d Dividing the ^^sin.^i ^^ =0. -| The value of r in this equation is the value of a the chord becomes 0, the value of r in the last equationbecomes 0 also, and then A^ h ;+-B* a Or ——- ,A-b a result corresponding to Prop. VI, as it ought to do, becausethe raditis vector then becomes tangent to the curve. Scholium 3. When P is placed at the extremity of the majoraxis on the right, then , , a=A, and h:= values substituted in the general equation will reduce itto B^r^+^B^Ar^O, which gives r=0, and r^—2,A, obviously true results. When F is placed at either foci, then a=JA^—B^=Cy and5=0. These values substituted, and we shall have It is difficult to deduce the values of r from this we adopt a more simple method. Let F be the focus, and FP anyradius, and put the angle PFD=:^v. By Prop. I, of the ellipse, we learnthat FP-. (1). an equation in which c=:^JA^—B^, and X any variable distance CD. Take the triangle PDF, and by trigonometry we have 1 \ r \ \ : c-\X. Whence a?=—c. This value of x placed in (1), will give A I cr COS. t;—c^r—ji-f- 64 ANALYTICAL GEOMETRY. WhenceOr (A—c )r=A^—c^ r=. A—c This equation will correspond to all points in the curve bygiving to ; all possible values from 1 to —1. Hence, thegreatest 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 astron
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