Vectors not separated by dots are fused into scalar products, as (be), (bg), etc., and here of course the order is irrelevant; but it must be carefully preserved in the resulting dyads, such as a. f, not f.a (unless a, f are collinear). Apart from this precaution, the multiplication of dyadics is as easy and convenient as the common multiplication of polynomials, and it will be found to render inestimable services in the treatment of many geometrical and physical, especially optical, problems. Some illustrations of the latter kind will be found in the "Simplified Method, etc.," mentioned before. The final result of such multiplications of two or more polynomials will be a polynomial of dyads, say A.B+C.D+E.F+G.H+ etc.; but since each of these antecedents and consequents can be expressed in the form xa+yb + zc, where a, b, c are any non-coplanar unit vectors, any such result can, in the first place, be reduced to a sum of nine dyads, viz. 11a.a+σb.b+σ33c.c+σ23b.c+σ32c.b+...+σb. a, and it can be proved that this can always be reduced by a proper choice of two orthogonal systems, i, j, k and 1, m, n, to the normal form (40), which is that of any, generally asymmetric, linear operator . Ultimately, the latter can with advantage be split into a symmetric operator w and the simple operator Vw, as in (32). 10. Hints on Differentiation of Vectors. The concepts of differentiation and integration as applied to vectors do not belong to the subject proper of this booklet, which is vector Algebra. Yet a few elementary remarks on the differentiation of vectorial expressions may be here added, as they are likely to be useful to some readers, and as they do by no means require much space. Let R be a variable vector. To have a possibly desirable picture think of R as the position-vector of a particle moving about in space, round a fixed origin. Let t be any independent scalar variable, say the time. Then, AR being the vector increment, i.e. the vector drawn from the position of the particle at the instant t to that at a later date t + At, the quotient AR/At will be a certain We may call it provisionally the average vector-velocity of the particle. If this quotient (a vector) tends, with indefinitely decreasing At, to some definite limit, definite both in size. and direction, we call this limit-vector the derivative or the fluxion of R with respect to t (or the vector velocity of the dR particle), and denote it by or R. In short symbols dt This vector will, in our illustration, be tangential to the orbit of the particle, and its tensor will represent the particle's speed. From this definition it follows at once that where R, S are any vector functions of the variable t. r be the unit of R, so that R=Rr, we have of course R=Rr + Rr. And, if Again, since the scalar product of two vectors is distributive, so that A(RS) =RAS+SAR plus terms of higher order, we have In particular, if r be a unit vector, so that r2=1, we have, by differentiating the latter condition, rr=0, so that rr, which is also an obvious property. Similarly, for the vector product, which again is distributive, the only precaution being to preserve the order of the factors, or -if this be inverted-to change the sign of the product in question. In quite the same way we have and so forth. d dt AVBC=AVBC+AVBC+AVBC, Even the case of linear vector functions, such as R, does not call for lengthy explanations. If not only the operand R, but also the nature of the operator @ varies with t, we have |