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 146 - Art. 136) that the tangent to the polar conic at the point where any chord meets it passes through the intersection of the tangents to the cubic at the points where it is met by the same chord, and is the harmonic conjugate to the line joining their intersection to the point 0. 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...
Page 12 - ... 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. 23. And in like manner when any theorem is demonstrated by line-coordinates, this is also a demonstration of the correlative theorem...
Page 85 - ... 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 («-«)+!> 08-y) = 0 .................. (1).
Page 100 - PRA, the angle which the incident ray makes with the normal to the curve, and RBM—PRM, the angle which the refracted ray makes with the same normal ; hence the ratio RA : RM is also given. Now since...
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 183 - AM' = RM, and therefore P = AR- AM, or p = 2r secw — 2r cosw = 2r tan CD sinw; or, in rectangular coordinates, The origin is therefore a cusp, and 2r - x an asymptote meeting the curve at an infinitely distant point of inflexion. Newton has given the following elegant construction for the description of this curve by continuous motion: A right angle has the side GF of fixed length, the point Amoves along the fixed line CI, while the side GH passes through the fixed point E; a pencil at the middle...
Page 312 - In like manner, through any point on a circular cubic or bicircular qnartic can be described nine circles elsewhere osculating the curve, and of these circles three will be real and their points of contact will lie on a circle passing through the given point. Ex. 3. " The feet of the perpendiculars on the sides of a triangle from any point on the circumscribing circle lie in one right line.
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...
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.

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