ON A GENERAL CENTRE OF APPLIED FORCES

By William Rowan Hamilton

(Proceedings of the Royal Irish Academy, 8 (1864), p. 394.)

Edited by David R. Wilkins

On a general Centre of applied Forces

Sir William Rowan Hamilton

[Read June 22nd, 1863.]

[Proceedings of the Royal Irish Academy, vol. viii (1864), p. 394.]

Sir W. R. Hamilton wishes a note to be preserved in the Proceedings of the Royal Irish Academy, that on recently reconsidering an application of Quaternions to the Statics of a Solid Body, some account of which was laid before the Academy many years ago (see the Proceedings for December 1845), he has been led to perceive the theoretical (and to suspect the practical) existence of a certain Central Point for every system of applied forces, not reducible to a couple, nor to zero: which generally new point, for the case of parallel forces, coincides with their well-known centre.

An applied force AB, acting at a point A, being said to have a quaternion moment, equal to the quaternion product OA . AB, with respect to any assumed point O, the sum of all such moments, or the quaternion, Q = Σ(OA . AB) = OA . AB + OA′ . A′B′ + &c., is called the total quaternion moment of the applied system with respect to the same point O.

This total moment Q varies generally with the point to which it is referred; and there is one point C, or one position of O, for which the condition

TQ = a minimum,

is satisfied, with the exceptions (of couple and equilibrium) above alluded to.

It is this point C, which Sir W. R. H. proposes to call generally the Centre of a System of Applied Forces.

In the most general case of such a system, he finds it to be situated on the Central Axis, the minimum TQ representing then what was called by Poinsot the Energy of the Central Couple.

For the less general case of an unique resultant force, the quaternion Q reduces itself to zero at the new Central Point C, which is now situated on the resultant, and determines its line of application.