Unpublished paper, 'On Clairaut's theorem and some matters connected with it' by Matthew Collins Collins begins his investigations by proving the existence of principal axes for any point of a body, which he makes to depend on the existence of principal axes of an auxiliary ellipsoid (Poinsot’s central one) having its centre at the given point, and such that any semidiameter of it is reciprocally proportional to the radius of gyration of the body about that semidiameter. He afterwards employs another ellipsoid (called McCullagh’s ellipsoid of inertia) concentric to the former and reciprocal to it, which admirably suits and facilitates the remainder of his investigations, and whose characteristic property is this, that it gives the radius of gyration itself (and not its reciprocal, as in Poinsot’s) about any semidiameter of it, the radius of gyration being in fact equal to the portion of that semidiameter between the centre and a tangent plane perpendicular to it. Subject: Mathematics / Geometry Received 2 May 1853. Communicated by Samuel Hunter Christie. Written by Collins at Upper Pitt Street , Liverpool [England]. Whilst the Royal Society declined to publish this paper in full, an abstract of the paper was published in volume 6 of Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London [later Proceedings of the Royal Society] as 'On Clairaut’s theorem and subjects connected with it'. December 1852