Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/195

Rh which lends itself so well to such broader treatment, is worthy of it. Of course, when vectors are treated by the methods of ordinary algebra, linear vector operators will naturally be treated by the same methods, but in an algebra formed for the sake of expressing the relations between vectors, and in which vectors are treated as multiple quantities, it would seem an incongruity not to apply the methods of multiple algebra also to the linear vector operator.

The dyadic is practically the linear vector operator regarded as quantity. More exactly it is the multiple quantity of the ninth order which affords various operators according to the way in which it is applied. I will not venture to say what ought to be included in a treatise on quaternions, in which, of course, a good many subjects would have claims prior to the linear vector operator; but for the purposes of my pamphlet, in which the linear vector operator is one of the most important topics, I cannot but regard a treatment like that in Hamilton's Lectures, or Elements, as wholly inadequate on the formal side. To show what I mean, I have only to compare Hamilton's treatment of the quaternion and of the linear vector operator with respect to notations. Since quaternions have been identified with matrices, while the linear vector operator evidently belongs to that class of multiple quantities, it seems unreasonable to refuse to the one those notations which we grant to the other. Thus, if the quaternionist has $$e^{q}, \log q, \sin q, \cos q,$$ why should not the vector analyst have $$e^{\Phi}, \log \Phi, \sin \Phi, \cos \Phi,$$ where $$\Phi$$ represents a linear vector operator? I suppose the latter are at least as useful to the physicist. I mention these notations first, because here the analogy is most evident. But there are other cases far more important, because more elementary, in which the analogy is not so near the surface, and therefore the difference in Hamilton's treatment of the two kinds of multiple quantity not so evident. We have, for example, the tensor of the quaternion, which has the important property represented by the equation: $$\text{T}(qr) = \text{T}q\text{T}r.$$

There is a scalar quantity related to the linear vector operator, which I have represented by the notation $$\left\vert \Phi \right\vert$$ and called the determinant of $$\Phi.$$ It is in fact the determinant of the matrix by which $$\Phi$$ may be represented, just as the square of the tensor of $$q$$ (sometimes called the norm of $$q$$) is the determinant of the matrix by which $$q$$ may be represented. It may also be defined as the product of the latent roots of $$\Phi,$$ just as the square of the tensor of $$q$$ might be defined as the product of the latent roots of $$q.$$ Again, it has the property represented by the equation which corresponds exactly with the preceding equation with both sides squared.