Mathematical Questions and Solutions

F. Hodgson., 1866

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Pàgina 47 - ... inflectional tangents of the Lemniscate ; in fact, they are the same lines. 2. Let s,f, (Fig. 1) be the inverse points of the foci S, F, and p the inverse of P ; then by similar triangles, we have sp : PO = sp : so, FP : PO =fp :fO ., but SP . FP = PO2, hence sp .fp = so .fo ; hence the lemniscate is the locus of the vertex of a triangle, whose base is given, and the rectangle under whose sides is equal to the square on half the base. 3. The rectangle contained by the intercepts made by a tangent...
Pàgina 39 - BC est conpee par les tangentes issues de /'. [Professor CATLET remarks that if ABCD is the perspective representation of a square, then the ellipse is the perspective representation of the inscribed circle ; the theorem gives eight points and the tangent at each of them ; and the ellipse may therefore be drawn by hand with an accuracy quite sufficient for practical purposes. The demonstration is immediate, by treating the figure as a perspective representation : the gist of the theorem is the very...
Pàgina 115 - SQP' are similar. Hence also SP : SQ :: SQ : SP' or SP . SP'=SQ2 QED PROP. 11.— THEOREM. 42. The circle circumscribing the triangle formed by any three tangents to a parabola passes through the focus. Let TRB/ be the triangle ; P, Q, P' the points of contact of the tangents.
Pàgina 109 - Sa-(/3+v+«) these then, with three similar pairs, express the eight roots as required.] 1823. (Proposed by WK CLIFFORD.) — The conicoids which pass through six fixed points in space, intersect any plane in a series of conies having a common self-conjugate quadrilateral. Any four conies have a common self-conjugate quadrilateral.
Pàgina 46 - Let a, t, c be the inclinations of the line (A, B, C) to the sides a, b, c. In any triangle, the product of a side and the sines of the adjacent angles is equal to the product of the perpendicular upon it from the opposite angle and the sine of that angle. Applying this to the triangles Aef, Bfd, Cde, we have...
Pàgina 56 - Prove that the characteristics (see Question 1573) of a system of conies, satisfying four conditions, remain unaltered when, in place of passing through a given point, each conic is required to divide a given finite segment harmonically.
Pàgina 75 - M ab + be + ca ~ a + b + c' which is evidently in the same straight line with O and the point (iv). Hence, in the projection, the centre of the nine-point circle is in the same straight line with the centre of the given circle and the intersection of the perpendiculars. 6. It will be seen that the same locus (iii) will be obtained if we Write...

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