A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections

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Hodges & Smith, 1852 - Curves, Algebraic - 316 pages
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Page 5 - The line joining the middle points of two sides of a triangle is parallel to the third side and equal to half of the third side.
Page 206 - This curve is generated by the motion of a point on the circumference of a circle which rolls along a right fcline.
Page 204 - ... a curve having the origin for a double point, and the two circular points at infinity for ordinary double points. As a generalization of the ovals of Cassini, we might seek the locus of a point, the product of whose distances from m given points shall be constant; and...
Page 118 - The angle which the incident ray makes with the normal to the surface...
Page 127 - ... where r, s, t are the distances of any point on the curve from the three foci. We shall enter into more detail on this subject in the next Chapter. Ex. 4. Equations of the form just mentioned include many well-known curves of the fourth degree. For example, the ovals „ of Cassini are the locus of the vertex of a triangle of which the base and rectangle under sides are given. Taking the middle point of the base 2a for origin, the given rectangle being ab, the equation is O 2 + f + a 2 - 2cui)...
Page 225 - Lumine emanans eidem 6{i6<3ios existit, qualiscumque adumbratio. Aut, si' mavis, quia curva nostra mirabilis in ipsa mutatione semper sibi constantissime manet similis et...
Page 136 - And again, the cube of the distance of any point of the curve from, this line is in a constant ratio to the product of its distances from the three tangents at the points of inflexion.
Page 109 - 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...
Page 225 - Cum autem ob proprietatem tam singularem tamque admirabilem mire mihi placeat Spira haec mirabilis, sic ut ejus contemplatione satiari vix queam; cogitavi, illam ad varias res symbolice repraesentandas non inconcinne adhiberi posse. Quoniam enim semper sibi similem...
Page 136 - F, and the product of the distances of any point of the curve from the three asymptotes is in a constant ratio to its distance from the line F.

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