Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/196

180 There is another scalar quantity connected with the quaternion and represented by the notation $$\text{S}q.$$ It has the important property expressed by the equation, and so for products of any number of quaternions, in which the cyclic order remains unchanged. In the theory of the linear vector operator there is an important quantity which I have represented by the notation $$\Phi_{\text{S}},$$ and which has the property represented by the equation where the number of the factors is as before immaterial. $$\Phi_{\text{S}}$$ may be defined as the sum of the latent roots of $$\Phi,$$ just as $$2\text{S}q$$ may be defined as the sum of the latent roots of $$q.$$

The analogy of these notations may be further iUustrated by comparing the equations and  I do not see why it is not as reasonable for the vector analyst to have notations like $$\left\vert \Phi \right\vert$$ and $$\Phi_{\text{S}},$$ as for the quatemionist to have the notations $$\text{T}q$$ and $$\text{S}q.$$

This is of course an argumentum, ad quaternionisten. I do not pretend that it gives the reason why I used these notations, for the identification of the quaternion with a matrix was, I think, unknown to me when I wrote my pamphlet. The real justification of the notations $$\left\vert \Phi \right\vert$$ and $$\Phi_{\text{S}}$$ is that they express functions of the linear vector operator quâ quantity, which physicists and others have continually occasion to use. And this justification applies to other notations which may not have their analogues in quaternions. Thus I have used $$\Phi_{\times}$$ to express a vector so important in the theory of the linear vector operator, that it can hardly be neglected in any treatment of the subject. It is described, for example, in treatises as different as Thomson and Tait's Natural Philosophy and Kelland and Tait's Quaternions, In the former treatise the components of the vector are, of course, given in terms of the elements of the linear vector operator, which is in accordance with the method of the treatise. In the latter treatise the vector is expressed by As this supposes the linear vector operator to be given not by a single letter, but by several vectors, it must be regarded as entirely inadequate by any one who wishes to treat the subject in the spirit of multiple algebra, i.e. to use a single letter to represent the linear vector operator.

But my critic does not like the notations $$\left\vert \Phi \right\vert, \Phi_{\text{S}}, \Phi_{\times}.$$ His ridicule,