Symmetrical Solution of the Ellipse-Glissette Elimination Problem

Extract

The publication of Mr Nanson's paper on the glissette elimination problem (Proc. Roy. Soc., vol. xxii. p. 158) has led me to make a further investigation of this interesting point, with the result that the general eliminant is expressed in the form of a single symmetrical bordered determinant. I shall first give the solution for the case where the tracing-point is on the axis of the ellipse, and then extend it to the general case.

I. Case of the Tracing-point on the Elliptic Axis.

The equations for the centre of the generating ellipse as tracing-point are—

For a point in the line of the major axis, if r be the distance of the tracing-point from the centre, we have

whence, by squaring each side,

By addition,