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—
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0370164600051312/resource/name/S0370164600051312_eqnU1.gif?pub-status=live)
For a point in the line of the major axis, if r be the distance of the tracing-point from the centre, we have
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0370164600051312/resource/name/S0370164600051312_eqnU2.gif?pub-status=live)
whence, by squaring each side,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0370164600051312/resource/name/S0370164600051312_eqnU3.gif?pub-status=live)
By addition,
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0370164600051312/resource/name/S0370164600051312_eqnU4.gif?pub-status=live)