A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections
Hodges, Foster and Figgis, 1873 - Conic sections - 379 pages
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A Treatise on the Higher Plane Curves - Intended as a Sequel to a Treatise ...
No preview available - 2010
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already angle axis becomes bitangents branches called centre circle coefficients coincide common condition conic considered constant contain coordinates corresponding counts cubic cusp degree denote described determine differential distances double point drawn eliminating envelope equal equation evidently evolute example expressed factor figure fixed points foci follows four functions give given given points Hence Hessian higher infinity intersection invariant line joining locus manner meets the curve method multiple point node normal obtained origin pair parallel parameters pass perpendicular points of contact points of inflexion polar polar conic pole position problem proved quartic radius ratio reciprocal reduced regard relation represents respectively result right line satisfy seen sides substituting tangent theorem theory third touch triangle values vanish write written
Page 106 - ... 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 59 - ... we shall call for shortness p, 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. 38). The normal then being a perpendicular to this at the point xy} has for its equation («-*)+!> OS-y) = 0 ................... (1).
Page 65 - ... of a 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 91 - O. 170. Let us now consider more particularly the case where 0 is a point of inflexion. It was shewn (Art. 74) that the polar conic of a point of inflexion breaks up into two right lines, one of them being the tangent at the point. And the same thing would appear from the equation of the polar conic of the origin just given. For, in order that the origin should be a point of inflexion and the axis of y the tangent at it, we must have (see Art. 46) A = 0, .5 = 0, D = 0, when the equation of the polar...
Page 56 - This equation represents the locus of a point, such that its polars with respect to U and F intersect on the assumed line. Now at a point common to U and F, the polars are the two tangents intersecting in the common point ; there are, therefore, plainly only two cases in which a point common to U and F can lie also on v...
Page 229 - P'Q? . . . = 0 ; the number and distribution of the summits is not arbitrary, but is regulated by laws arising from the consideration of the penultimate curve, and there are of course for any given value of n various forms of degenerate curve, according to the different ultimate forms P*QP ...= 0, and to the number and distribution of the summits on the different component curves.
Page 227 - ... 0, but we have to consider how the point-pair can be represented in point-coordinates : an equation a? = 0, is no adequate representation of the point-pair, but merely represents (as a twofold or twice repeated line) the line joining the two points of the point-pair, all traces of the points themselves being lost in this representation : and it is to be noticed that the conic, or two-fold line, a?
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 186 - ... will be The tractrix is a particular case of the general problem of equi-tangential curves, where it is required to find a curve such that the intercept on the tangent between the curve and a fixed directrix shall be constant. 318. The problem of curves of pursuit was first presented in the form — To find the path described by a dog which runs to overtake its master. It may be stated mathematically as follows: The point A describes a known curve, and it is required to find the curve described...