This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1835 Excerpt: ...to which it belongs. If FP be produced to P', the tangent at P' must meet that through P on the directrix, as in the ellipse, and for a like reason. Comparing the expression for the distance of any point in the curve from the directrix, viz. x =--with ex--A, the distance of the e e same point from the focus, we find ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1835 Excerpt: ...to which it belongs. If FP be produced to P', the tangent at P' must meet that through P on the directrix, as in the ellipse, and for a like reason. Comparing the expression for the distance of any point in the curve from the directrix, viz. x =--with ex--A, the distance of the e e same point from the focus, we find that their ratio is constant, viz. as 1 to e; e being y 1, the distance of any point from the focus is greater than its distance from the directrix. From what has now been said, we may define the directrix of either of the three curves to be the locus of the extremities of the polar subtangents, or else as the locus of the intersection of pairs of tangents at the extremities of the focal clwrds. In $ie parabola, the distance of any point from the focus is equal to its distance from the directrix, but in the ellipse it is less, and in the hyperbola greater. (117.) By referring to the value of p, the principal parameter in each of the three curves, we find that in the ellipse p--A (1--e2), in the hyperbola p = A (e2--1), and in the parabola p = 2m; hence, by substituting p for these values in the polar equations of the respective curves, they will then all become comprehended in a single equation, viz. r =, --. If we suppose r produced to meet the l-f-ecos.u curve again below the principal axis, we shall have the value of r', the radius vector of this second point, by changing oo into 180" + that is r'----. Hence, for any focal chord, we have 1--e cos. u r4-i/ = f.--=--also rr'-. 5--consequently 1--e' cos. to 1--e cos.. u (r-f-r') p--4rr'...-= p; hence focal chords are to each other as the rectangles of the parts into which they are divided by the focus, likewise half the principal parameter is always an harmonical mean between the parts into ...
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Add this copy of The Elements of Analytical Geometry to cart. $20.57, new condition, Sold by Ingram Customer Returns Center rated 5.0 out of 5 stars, ships from NV, USA, published 2022 by Legare Street Press.
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Add this copy of The Elements of Analytical Geometry to cart. $32.92, new condition, Sold by Ria Christie Books rated 5.0 out of 5 stars, ships from Uxbridge, MIDDLESEX, UNITED KINGDOM, published 2022 by Legare Street Press.
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