Mathematical Questions and Solutions, from the "Educational Times": With Many Papers and Solutions in Addition to Those Published in the "Educational Times", Volumen7W. J. C. Miller Hodgson, 1867 |
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Términos y frases comunes
angle axes b₁ bisected Cartesian centre chance of falling chord circular points circumscribing coaxal concyclic points coordinates cubic curve curve DALE diameter distance drawn at random ellipse envelope equally probable foci focus four given points four points given line GODFRAY gonal harmonic conjugate harmonically hence hyperbola inscribed line drawn line joining line whose chance MCCORMICK middle point nine-point circle P₁ pair parabolas parallel pass perpendiculars plane points at infinity points be taken Proposed by Professor Proposed by W. K. prove quadric question radical axis radius random line random point Reprint respectively sides BC sin² Solution STEPHEN WATSON straight line suppose taken at random tangents tetrahedron theorem three points TOMLINSON triangle ABC trilinear coordinates values variable W. K. CLIFFORD whence
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Página xv - Fifteen young ladies in a school walk out three abreast for seven days in succession : it is required to arrange them daily, so that no two shall walk twice abreast...
Página 99 - From a point in the circumference of a circular field a projectile is thrown at random with a given velocity which is such that the diameter of the field is equal to the greatest range of the projectile : find the chance of its falling within the field.
Página 84 - I am touching on need much ampler developments than the limits of a Note permit ; so that much must be left to the sagacity of the reader. The expression " at random" has in common language a very clear and definite meaning ; one which cannot be better conveyed than by Mr. Wilson's definition, — " according to no law ;" and in this sense alone I mean to use it in this Note.
Página 47 - D of four circles (A), (B), (C), (U) cutting each other at right angles in a plane, is perpendicular to the line joining the other two; any circle (A) cuts the sides of the triangle BCD harmonically, and the square of its radius varies inversely as the area of the triangle BCD. The sum of the squares of the radii is equal to the square of the diameter of the equal circles circumscribing the triangles, and the sum of the inverse squares is nothing. 2. If the circle described on AB as a diameter be...
Página 96 - If a conic pass through the centres of the four circles which touch the sides of a triangle it must be a rectangular hyperbola, and its centre will lie on the circumscribed circle of the triangle.
Página x - If A, B, C, D, E be five points on a circle, the consecutive intersections of the nine-point circles of the triangles ABC, BCD, CDE, DEA, EAB lie on another circle whose radius is one half that of the first. (16) In triangle ABC the circles described on AG, B'C', EC as diameters are coaxal. If G...
Página xv - CAA3B4; prove that two circles and another point may be taken arbitrarily, and that the circles abc meet the circles def in six new points which lie on the circumference of another circle.
Página 75 - Cartesian ovals which can be drawn through four given concyelic points is identical with the locus of the foci of the conies which pass through the same four points; viz., the two circular cubics of which those points are foci. This extremely remarkable theorem states an apparent absurdity. As six conditions are required to determine a Cartesian oval, if we are given four points on the curve, we might apparently take any point whatever for a focus; and in fact this is true in general, provided the...
Página 34 - P = + s, ^/Q = + s, or the equation in* is . (s, 1) (±s, 1) (+ », 1) = 0; that is, the equation of the twelfth order breaks up into four equations each of the third order. The geometrical theory may also be further developed. In fact, assuming on each of the three lines respectively a certain sense as positive (and thus isolating a set of three solutions) the construction is, on the three lines, from the points A', B', C' respectively, measure off the distances A'A^ B'B = C'C=*.
Página 50 - The latter, therefore, are parallel to the axes of the parabolas, since both are perpendicular to the same two lines. [Mr. TOWNSEND gives the proof as follows : — " The property appears immediately from the two considerations that, if from any two points on a parabola perpendiculars be drawn to the axis, the distance between their feet is equal to the difference of the focal radii of the points, and that, if from the two foci of an hyperbola perpendiculars be drawn to either asymptote, the distance...