Reference numberAP/31/3
TitleUnpublished paper, 'On the application of the theory of elliptic functions and of the properties of surfaces of the second order to the rotation of a rigid body round a fixed point' by James Booth
DescriptionBooth provides an overview of existing research into the motion of a body round a fixed point, which, when free from the action of accelerating forces, is reduced to the motion of a certain ellipsoid whose centre is fixed, and which rolls without sliding on a plane fixed in space. He remarks upon the identity which exists between the formulae for finding the position of the principal axes of a body and those for determining the symmetrical diameters of an ellipsoid; and further observes that the expression for the perpendicular from the centre on a tangent plane to an ellipsoid, in terms of the cosines of the angles which it makes with the axes, is precisely the same in form as that which gives the value of the moment of inertia round a line passing through the origin. Guided by this analogy, he is led to assume an ellipsoid the squares of whose axes should be directly proportional to the moments of inertia round the coinciding principal axes of the body. In the first section of the paper, Booth establishes such properties as he has subsequently occasion to refer to, of cones of the second degree, and of the curves of double curvature in which these surfaces may be intersected by concentric spheres, some, of which he believes will not be found in any published treatise on the subject. He considers that he has been so fortunate as to be the first to obtain the true representative curve of elliptic functions of the first order. It is shown that any spherical conic section, the tangents of whose principal semiarcs are the ordinates of an equilateral hyperbola whose transverse semiaxis is 1, may be rectified by an elliptic function of the first order, and the quadrature of such a curve may be effected by a function of the same order, when the cotangents of the halves of the principal arcs are the ordinates of the same equilateral hyperbola.

Annotations in pencil throughout.

Subject: Mathematics / Geometry

Received 30 November 1848. Read 8 February 1849.

Whilst the Royal Society declined to publish this paper in full, an abstract of the paper was published in volume 5 of Abstracts of the Papers Printed in the Philosophical Transactions of the Royal Society of London [later Proceedings of the Royal Society] as 'On the application of the theory of elliptic functions to the rotation of a rigid body round a fixed point'.

A version of this paper was published by Booth as a book: Booth, James. The Theory of Elliptic Integrals, and the Properties of Surfaces of the Second Order, Applied to the Investigation of the Motion of a Body Round a Fixed Point (London: George Bell & Sons, 1851).
Physical descriptionInk and graphite pencil on paper
Access statusOpen
Related materialDOI: 10.1098/rspl.1843.0178
Fellows associated with this archive
NA7648Booth; James (1806 - 1878)1806 - 1878
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