Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/192

Or, if the problem and solution had been written thus: Bequired $\omega$ in terms of $\nabla \omega$ when ${\text{S}}\nabla \omega =0.$

Solution: $\omega =\nabla \nabla ^{-2}\nabla \omega =\nabla ^{-1}\nabla \omega .$

My critic has himself given more than one example of the unfitness of the inverse Nabla for the exact expression of thought. For example, when he says that I have taken "eight distinct steps to prove two equations, which are special cases of

$\nabla ^{-2}\nabla ^{2}u=u,$"

I do not quite know what he means. If he means that I have taken eight steps to prove Poisson's Equation (which certainly is not expressed by the equation cited, although it may perhaps be associated with it in some minds), I will only say that my proof is not very long, especially as I have aimed at greater rigor than is usually thought necessary. I cannot, however, compare my demonstration with that of quaternionic writers, as I have not been able (doubtless on account of insufficient search) to find any such.

To show how little foundation there is for the charge that the deficiencies of my system require to be pieced out by these integral operators, I need only say that if I wished to economise operators I might give up ${\text{New, Lap,}}$ and ${\text{Max,}}$ writing for them $\nabla {\text{Pot, }}\nabla \times {\text{Pot, }}$ and $\nabla .{\text{Pot, }}$ and if I wished further to economise in what costs so little, I could give up the potential also by using the notation $(\nabla .\nabla )^{-1}$ or $\nabla ^{-2}.$ That is, I could have used this notation without greater sacrifice of precision than quatemionic writers seem to be willing to make. I much prefer, however, to avoid these inverse operators as essentially indefinite.

Nevertheless—although my critic has greatly obscured the subject by ridiculing operators, which I beg leave to maintain are not worthy of ridicule, and by thoughtlessly asserting that it was necessary for me to use them, whereas they are only necessary for me in the sense in which something of the kind is necessary for the quaternionist also if he would use a notation irreproachable on the score of exactness—I desire to be perfectly candid. I do not wish to deny that the relations connected with these notations appear a little more simple in the quaternionic form. I had, indeed, this subject principally in mind when I said two years ago in Nature (vol. xliii, p. 512) [this vol. p. 158]: "There are a few formulæ in which there is a trifling gain in compactness in the use of the quaternion." Let us see exactly how much this advantage amounts to.

There is nothing which the most rigid quaternionist need object to in the notation for the potential, or indeed for the Newtonian. These represent respectively the operations by which the potential or the force of gravitation is calculated from the density of matter. A quaternionist would, however, apply the operator $New$ not only to a