He gave, in fact, the theory of what in Hamilton's system is called Composition of Vectors in one plane i.e.
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.
For if 0, A, B be any three points, and m, n any scalar quantities, we have in vectors m.~+n.~=(m+n)O~, (I)
Of the vectors AB and AC (or BD), and of their sum AD, on a line perpendicular to AO, this is obvious.
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.
Since the projection of a vector- sum is the sum of the projections of the several vectors, the equation (2) gives if x be the projection of 0G.
Newtons Second Law asserts that change of momentum is equal to the impulse; this is a statement as to equality of vectors and so implies identity of direction as well as of magnitude.
Kinetics of a System of Discrete Particles.The momenta of the several particles constitute a system of localized vectors which, for purposes of resolving and taking moments, may be reduced like a system of forces in statics (~ 8).
Fixed point 0 we draw vectors 0V1, 0V2.
Beare (Oxford, 1890), and a discussion of the subject of reciprocal figures from the special point of view of the engineering studenl is given in Vectors and Rotors by Henrici and Turner (London, 1903) See also above under Theoretical Mechanics, Part I.
Then assuming that the acceleration of one point of a particuar link of the mechanism is known together with the corresponding configuration of the mechanism, the two Vectors Ac and ct can be drawn.
The conditions (c) may thus be expressed: first, that the sum of the vectors Wr must form a closed polygon, and, second, that the sum of the vectors War must form a closed polygon.
In fact, whatever theory of light be adopted, there are two vectors to be considered, that are at right angles to one another and connected by purely geometrical relations.
In the main such investigations have only an academic interest, as, whatever theory of light be adopted, we have to deal with two vectors that are parallel and perpendicular respectively to the plane of polarization.
203) show that chemical action is to be referred to the latter of these vectors, but whether Fresnel's or Neumann's hypothesis be correct is only to be decided when we know if it be the mean kinetic energy or the mean potential energy that determines chemical action.
Thus every quaternion may be written in the form q = Sq+Vq, where either Sq or Vq may separately vanish; so that ordinary algebraic quantities (or scalars, as we shall call them) and pure vectors may each be regarded as special cases of quaternions.
When Clerk Maxwell pointed out the way to the common origin of optical and electrical phenomena, these equations naturally came to repose on an electric basis, the connexion having been first definitely exhibited by FitzGerald in 1878; and according as the independent variable was one or other of the vectors which represent electric force, magnetic force or electric polarity, they took the form appropriate to one or other of the elastic theories above mentioned.