The Journal of the Indian Mathematical Society

N. Durairajan

Abstract

The triangle ABC formed b? the three fixed asymptotes of the net of plane cubic curves is taken as the triangle of reference. PQR is any line cutting the curve in p, q, r and the sides in P, Q, R. It is known that the mean centre of P, Q> R, coincides with that of p, q, r.

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N. Durairajan

Abstract

A curve of class three having the line at infinity as a bitangent in two perpendicular directions has a straight line for its orthoptic locus. This curve has been identified in this paper as the negative pedal of a rectangular hyperbola with respect to a point on itself. The Cartesian equation of the curve is obtained and the curve is shown to possess properties remarkably analogous to those of the parabolat a fact, first pointed out by Mr. A. Narasinga Rao The curve has been named the harmonic biparabola The name biparabola has been chosen as being shorter ihan the term " two-fold parabola " used by Clifford,↓ who gives also a method of generation Other methods of generating the curve are also indicated in the paper.

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N. Durairajan

Abstract

The object of this paper is to indicate the part played by the concept 'focus' in the geometry of the Gauss plane. The relation between focal properties and complex geometry is a reciprocal one. On the one hand, the method of complex representation may be utilised to elucidate the properties of real foci of curves; this is what is done in para. 1 of the paper, where a general theorem is stated, which gives immediately the position of the real foci of a curve given by a real areal tangential equation.

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N. Durairajan

Abstract

The tetrahedron in which the products of opposite edges are equal has some interesting properties Professor Wilkinson has discussed some of these analytically on pp. 115-7 of the present volume of the J . l . M.S. Geometrcal pischolar_main are also simple enoagh as may be gathered from the following note.

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