He gave, in fact, the theory of what in Hamilton's system is called Composition of Vectors in one plane i.e.
The two forces at B will cancel, and we are left with a couple of moment P.AC in the plane AC. If we draw three vectors to represent these three couples, they will be perpendicular and proportional to the respective sides of the triangle ABC; hence the third vector is the geometric sum of the other two.
In octonions the analogue of Hamilton's vector is localized to the extent of being confined to an indefinitely long axis parallel to itself, and is called a rotor; if p is a rotor then wp is parallel and equal to p, and, like Hamilton's vector, wp is not localized; wp is therefore called a vector, though it differs from Hamilton's vector in that the product of any two such vectors wp and coo- is zero because w 2 =o.
In modern language, forces are compounded by vector-addition; thus, if we draw in succession vectors ~--~
-, HK be vectors representing the given forces, the resultant will be given by AK.