Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/193

scalar, as I have done, but to a vector also. The vector part of ${\text{New }}\omega$ (construed in the quaternionic sense) would be exactly what I have represented by ${\text{Lap }}\omega ,$ and the scalar part, taken negatively, would be exactly what I have represented by ${\text{Max }}\omega .$ The quaternionist has here a slight economy in notations, which is of less importance, since all the operators—${\text{New, Lap, Max,}}$—may be expressed without ambiguity in terms of the potential, which is therefore the only one necessary for the exact expression of thought.

But what are the formulas which it is necessary for one to remember who uses my notations? Evidently only those which contain the operator $Pot.$ For all the others are derived from these by the simple substitutions

${\begin{aligned}{\text{New}}&=\nabla {\text{Pot,}}\\{\text{Lap}}&=\nabla \times {\text{Pot,}}\\{\text{Max}}&=\nabla .{\text{Pot.}}\end{aligned}}$

Whether one is quatemionist or not, one must remember Poisson's Equation, which I write

$\nabla .\nabla {\text{Pot }}\omega =-4\pi \omega ,$

and in quaternionic might be written

$\nabla ^{2}{\text{Pot }}\omega =4\pi \omega .$

If $\omega$ is a vector, in using my equations one has also to remember the general formulæ,

$\nabla .\nabla \omega =\nabla \nabla .\omega -\nabla \times \nabla \times \omega$

which as applied to the present case may be united with the preceding in the three-membered equation,

$\nabla .\nabla {\text{Pot }}\omega =\nabla \nabla .{\text{Pot }}\omega -\nabla \times \nabla \times {\text{Pot }}\omega =-4\pi \omega .$

This single equation is absolutely all that there is to burden the memory of the student, except that the symbols of differentiation ($\nabla ,\nabla \times ,\nabla .$) may be placed indifferently before or after the symbol for the potential, and that if we choose we may substitute as above $New$ for $\nabla {\text{Pot,}}$ etc. Of course this gives a good many equations, which on account of the importance of the subject (as they might almost be said to give the mathematics of the electro-magnetic field) I have written out more in detail than might seem necessary. I have also called the attention of the student to many things, which perhaps he might be left to himself to see. Prof. Knott says that the quaternionist obtains similar equations by the simplest transformations. He has failed to observe that the same is true in my Vector Analysis, when once I have proved Poisson's Equation. Perhaps he takes his model of brevity from Prof. Tait, who simplifies the subject, I believe, in his treatise on Quaternions, by taking this theorem for granted.

Nevertheless, since I am forced so often to disagree with Prof. Knott, I am glad to agree with him when I can. He says in his