Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/174

The quantities written ${\text{S}}\nabla \omega$ and ${\text{V}}\nabla \omega$ where $\omega$ denotes a vector having values which vary in space, are of fundamental importance in physics. In quaternions these are derived from the quaternion $\nabla \omega$ by selecting respectively the scalar or the vector part. But the most simple and elementary definitions of ${\text{S}}\nabla \omega$ and ${\text{V}}\nabla \omega$ are quite independent of the conception of a quaternion, and the quaternion $\nabla \omega$ is scarcely used except in combination with the symbols ${\text{S}}$ and ${\text{V}},$ expressed or implied. There are a few formulæ in which there is a trifling gain in compactness in the use of the quaternion, but the gain is very trifling so far as I have observed, and generally, it seems to me, at the expense of perspicuity.

These considerations are sufficient, I think, to show that the position of the quatemionist is not the only one from which the subject of vector analysis may be viewed, and that a method which would be monstrous from one point of view, may be normal and inevitable from another.

Let us now pass to the subject of notations. I do not know wherein the notations of my pamphlet have any special resemblance to Grassmann's, although the point of view from which the pamphlet was written is certainly much nearer to his than to Hamilton's. But this a matter of minor consequence. It is more important to ask. What are the requisites of a good notation for the purposes of vector analysis? There is no difference of opinion about the representation of geometrical addition. When we come to functions having an analogy to multiplication, the products of the lengths of two vectors and the cosine of the angle which they include, from any point of view except that of the quatemionist, seems more simple than the same quantity taken negatively. Therefore we want a notation for what is expressed by $-{\text{S}}\alpha \beta ,$ rather than ${\text{S}}\alpha \beta ,$ in quaternions. Shall the symbol denoting this function be a letter or some other sign? and shall it precede the vectors or be placed between them? A little reflection will show, I think, that while we must often have recourse to letters to supplement the number of signs available for the expression of all kinds of operations, it is better that the symbols expressing the most fundamental and frequently recurring operations should not be letters, and that a sign between the vectors, and, as it were, uniting them, is better than a sign before them in a case having a formal analogy with multiplication. The case may be compared with that of addition, for which $\alpha +\beta$ is evidently more convenient than $\textstyle \sum \displaystyle (\alpha ,\beta )$ or $\textstyle \sum \displaystyle \alpha \beta$ would be. Similar considerations will apply to the function written in quaternions ${\text{V}}\alpha \beta .$ It would seem that we obtain the ne plus ultra of simplicity and convenience, if we express the two functions by uniting the vectors in each case with a sign suggestive of multiplication. The particular forms of the signs which