*Researches in Symbolical Physics. On the Translation of a Directed Magnitude as Symbolised by a Product. The Principles of Statics established symbolically*, by the Rev M O'Brien, M.A., late Fellow of Caius College, Cambridge, and Professor of Natural Philosophy and Astronomy in King's College, London. This paper, communicated by W A Miller, M.D., F.R.S. was received on 10 April 1851. The following is the abstract of this paper:

**Abstract. Researches in Symbolical Physics.**

In this communication the author (starting from the well-known theorem, that two sides of a triangle are equivalent to the third, when *direction*, as well as magnitude, is taken into account) proposes an elementary step in symbolization which consists in representing the *Translation of a Directed Magnitude* by a *Product*. Any magnitude which is drawn or points in a particular direction, such as a force, a velocity, a displacement, or any of those geometrical or physical quantities which we exhibit on paper by *arrows*, he calls a *directed magnitude*. By the *translation* of such a magnitude he means the removal of it from one position in space to another *without change of direction*.

**U** representing any directed magnitude and **u** any distance, the translation of **U** to any parallel position in space, in such wise that every point or element of **U** is caused to describe the distance **u**, is termed the *translation of* **U** *along* **u**.

This translation consists generally of two distinct changes, one the *lateral* shifting of the line of direction of **U**, and the other the motion of **U** *along* its line of direction. The former is called the *transverse effect*, the latter the *longitudinal effect* of the translation of **U** along **u**.

Both these effects are shown to be *products* of **U** and **u**; the transverse effect is represented by **uU**, and the longitudinal by **u.U**, inserting a dot between the factors in the latter for the sake of distinction.

The author then goes on to apply the principles established to the proof of the *Parallelogram of Forces*, and the determination of the effect of any set of forces on a rigid body. In doing this a remarkable symbolization of the *point of application*, as well as the direction and magnitude of a force, is obtained, namely, that the expression **(1 + u)U** represents a force **U** acting at a distance **u** from the origin.

The principles of statics are deduced with remarkable facility from the symbolical representation of the translation of a force along a given distance.