1. A quaternion r = Sr + Vr denotes the sum of a scalar and a vector, the former being an essentially undirected quantity. In many cases, however, and specially in the theory of screws, we have to deal with two co-directed quantities. In the usual notation, the components of a translation, λ, which are parallel to, and perpendicular to, a rotation, μ are represented respectively by the first and second terms in the identity λ = (Sλμ−1 + Vλμ−1)μ. The axis of the corresponding screw of pitch Sλμ−1 has the direction of μ., and Vλμ−1 is the perpendicular upon it from the origin. A certain advantage in point of unity would arise from taking Sr and TVr to represent respectively the magnitudes of the translation and the rotation in a screw whose axis passes through the origin, and has the direction of Vr. When such a use is made of a quaternion, it is necessary to attach a special symbol. Thus, Mr may be taken to denote the motor Sr·UVr + TVr·UVr whose axis passes through the origin and whose pitch is Sr/TVr.