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. 1854 Excerpt: ...that angle the same in each; if CB, C'B' intersect in D, and B' move up to B, then in the limit DC: DB:: AB: AC. From (fig. 74) C draw CE parallel to AB meeting B'C in E. Because AB ] AC = AB' + AC, BB' = CC; K 2. Define the circle of curvature at any point of a curve. If PQ be an arc, and QR a subtense, the chord of ...
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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. 1854 Excerpt: ...that angle the same in each; if CB, C'B' intersect in D, and B' move up to B, then in the limit DC: DB:: AB: AC. From (fig. 74) C draw CE parallel to AB meeting B'C in E. Because AB ] AC = AB' + AC, BB' = CC; K 2. Define the circle of curvature at any point of a curve. If PQ be an arc, and QR a subtense, the chord of the circle of curvature at P parallel to QR is equal to the limit of the third proportional to QR and PQ. Find the chord of curvature through the focus of an ellipse. EF is a chord of a given circle and 8 its middle point; construct the ellipse of which E is one point, S one focus, and the given circle the circle of curvature at E. SE.HE The chord of curvature (fig. 75) through the focus = 2--q--if Hbe the second focus and AC the semi-major axis. But in this case the chord is equal to 2SE. Hence HE= A C, and E is the extremity of the minor axis of the ellipse. Draw through E the chord EG making the same angle with the tangent at E that EF does. The middle point of this chord will be the second focus H, and the ellipse is constructed. 3. Shew that, in an orbit described under the action of a force tending to a fixed point, the velocity at any point is inversely proportional to the perpendicular from the centre of force on the tangent at that point. A body is describing a parabola under the action of a force which always tends to the focus, and a straight line is drawn from the focus perpendicular to the tangent, and proportional to the velocity, at any point; shew that the extremity of this straight line will lie in a certain circle. Draw SY (fig. 76) perpendicular from the focus S on the tangent at P. Produce it to Q, so that SQ bears a certain ratio to the velocity at P, and in the axis take a point B, such that SB bears the same ratio to the v...
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