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

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Hodges, Foster, and Figgis, 1879 - Curves, Algebraic - 395 pages
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Page 275 - This curve is generated by the motion of a point on the circumference of a circle which rolls along a right jline.
Page 83 - Art. 248) as the locus of the centres of curvature of the curve ; but the evolute may also be defined as the envelope of all the normals of the curve. For the circle of curvature is that which passes through three consecutive points of the curve, and its centre is the intersection of perpendiculars at the middle points of the sides of the triangle formed by the points. But the lines joining the first and second, and the second and third points, are two consecutive tangents to the curve ; and the...
Page 181 - ... inscribed in a circle. 514. Find the shortest distance between two circles which do not meet. 515. Two circles cut one another at a point A : it is required to draw through A a straight line so that the extreme length of it intercepted by the two circles may be equal to that of a given straight line. 516. If a polygon of an even number of sides be inscribed in a circle, the sum of the alternate angles together with two right angles is equal to as many right angles as the figure has sides. 517....
Page 12 - ... from it by the theory of reciprocal polars (or that of geometrical duality), viz. we do not demonstrate the first theorem and deduce from it the other, but we do at one and the same time demonstrate the two theorems; our (or, y, z) instead of meaning point-coordinates may mean line-coordinates, and the demonstration is in every step thereof a demonstration of the correlative theorem.
Page 100 - ... circle. Let a circle be described through A, the radiant point, and R, the point of incidence, to touch OR; then the point B is given, since OA. OB = OR*.
Page 85 - ... is to be found from the equation of the curve. For the tangent passes through the point xy, and makes with the axis of x an angle whose tangent is p (Art.
Page v - Algebra was still in its infancy, required extensive alterations in order to bring it up to the present state of the science; and...
Page 146 - J\ but the theorem of Newton's, which has hitherto guided us, is perhaps insufficient immediately to furnish a second relation. Such a relation, however, may be obtained by the following considerations. It is well known that the polar conic of a point of inflexion breaks up into two lines : one of these is the tangent at the point of inflexion, the other will be found to be the locus of the harmonic centres of the n - 1 points in which the curve is cut by a transversal through the point. Exactly...
Page 183 - The curve may also be defined as the locus of the foot of a perpendicular let fall from the vertex of a parabola upon a tangent. The problem of "duplicating the cube" is not taken up directly by Cantor.
Page 54 - Waring was the first who investigated the problem of the number of tangents which can be drawn from a given point to a curve of the n"t degree.

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