Mathematical Questions and Solutions, from the "Educational Times": With Many Papers and Solutions in Addition to Those Published in the "Educational Times", Volume 7
W. J. C. Miller
Hodgson, 1867 - Mathematics
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angle axis becomes centre chance chords circle circular common condition conic contain coordinates cubic curve DALE definition determined diameter direction distance divide double draw drawn ellipse envelope equal equation evidently expression falling fixed foci focus follows four four points given points gives harmonically Hence hyperbola infinity inscribed intersection joining length locus mean meet middle point obtain opposite pair parabolas parallel passing perpendicular plane position probability problem Professor Proposed prove question radical radius random line random point reference Reprint respectively result roots sides Solution space square straight line suppose taken at random tangent term theorem touching triangle ABC values variable whence
Page 17 - 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...
Page 86 - 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.
Page 49 - 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...
Page 98 - 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.
Page xii - 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...
Page 17 - 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.
Page 77 - 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...
Page 36 - 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=*.
Page 52 - 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...