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" The length of any arc of the evolute is equal to the difference of the radii of curvature at its extremities. For, draw any three consecutive normals to the original curve: let C be the point of intersection of the first and second, C... "
A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on ... - Page 109
by George Salmon - 1852 - 316 pages
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Report of the Board of Education

Bombay (India : State). Board of Education - Education - 1851 - 768 pages
...inflexion. 24. Show geometrically that the normals to a curve are tangents to the evolute. 25. Also that an arc of the evolute is equal to the difference of the radii of curvature at its extremities. 26. If an ellipse and circle intersect in four points, the lines joining the points of intersection...
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A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on ...

George Salmon - Conic sections - 1852 - 338 pages
...sign, and that as we pass such a point the concavity of the curve changes to convexity, and vice versd (see fig. p. 35). At a double point the radius of...at its extremities. For, draw any three consecutive norinals to the original curve : let C be the point of intersection of the first and second, C' of...
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A treatise on infinitesimal calculus, Volume 1

Bartholomew Price - Calculus - 1852
...with straight lines, whence they are said to be rectifiable ; for, from what has preceded, the length of the evolute is equal to the difference of the radii of curvature of the involute corresponding to its two extremities. Of this we subjoin some examples : Ex. 1. To...
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Mathematical Dictionary and Cyclopedia of Mathematical Science: Comprising ...

Charles Davies, William Guy Peck - Electronic book - 1855 - 592 pages
...the normals are more or less numerous. . This property, taken in connection with the property that any arc of the evolute is equal to the difference of the radii of curvature of the involute through its extremities, enables us to construct the involute when the evolute and...
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Report of the Board of Education

Bombay (India : State). Board of Education - Education - 1851 - 764 pages
...inflexion. 24. Show geometrically that the normals to a curve are tangents to the evolute. 25. Also that an arc of the evolute is equal to the difference of the radii of curvature at its extremities. 26. If an ellipse and circle intersect in four points, the lines joining the points of intersection...
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A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on ...

George Salmon, Arthur Cayley - Curves, Algebraic - 1873 - 379 pages
...own curvature at the point. At a cusp it will be found that the radius of curvature vanishes. 103. The length of any arc of the evolute is equal to the...three consecutive normals to the original curve : let G be the point of intersection of the first and second, G' of the second and third ; then since, ultimately,...
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The Elements of the Differential and Integral Calculus: With Numerous Examples

Donald Francis Campbell - Calculus - 1904 - 394 pages
...establish two theorems. Theorem I. Every normal to the involute is tangent to the evolute. Theorem II. The length of any arc of the evolute is equal to the difference between the lengths of the radii of curvature of the involute which pass through the extremities of...
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A Book of Curves

Edward Harrington Lockwood - Curves - 1967 - 286 pages
...TV is the centre of curvature for the original spiral at P. As shown on p. 84 for the cycloid, the arc of the evolute is equal to the difference of the radii of curvature at its end-points. The length of the new spiral from N to the pole (or as near to the pole as may be) is,...
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