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.