Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/191

Rh which I have iised by quoting a aentence addressed to the British Association a few years ago. The speaker was Lord Kelvin.

"Helmholtz first solved the problem—Given the spin in any case of liquid motion, to find the motion. His solution consists in finding the potentials of three ideal distributions of gravitational matter having densities respectively equal to $$1 / \pi$$ of the rectangular components of the given spin; and, regarding for a moment these potentials as rectangular components of velocity in a case of liquid motion, taking the spin in this motion as the velocity in the required motion" (Nature, vol. xxxviii, p. 569).

In the terms and notations of my pamphlet the problem and solution may be thus expressed:

Given the curl in any case of liquid motion—to find the motion.

The required velocity is $$1 / 4\pi$$ of the curl of the potential of the given curl.

Or, more briefly—The required velocity is $$\frac{1}{4\pi}$$ of the Laplacian of the given curl.

Or in purely analytical form—Required $$\omega$$ in terms of $$\nabla \times \omega,$$ when $$\nabla. \omega = 0.$$

Solution: $$\omega = 1 / 4 \pi \nabla \times \text{Pot } \nabla \times \omega = 1/4\pi \text{Lap } \nabla \times \omega.$$

(The Laplacian expresses the result of an operation like that by which magnetic force is calculated from electric currents distributed in space. This corresponds to the second form in which Helmholtz expressed his result.)

To show the incredible rashness of my critics, I will remark that these equations are among those of which it is said in the original paper (Proc. R.S.E., Session 1892–93, p. 225), "Gibbs gives a good many equations—theorems I suppose they ape at being." I may add that others of the equations thus characterized are associated with names not less distinguished than that of Helmholtz. But that to which I wish especially to call attention is that the terms and notations in question express exactly the notions which physicists want to use.

But we are told (Nature, vol. xlvii, p. 287) that these integrating operators ($$\text{Pot, Lap}$$) are best expressed as inverse functions of $$\nabla.$$ To see how utterly inadequate the Nabla would have been to express the idea, we have only to imagine the exclamation points which the members of the British Association would have looked at each other if the distinguished speaker had said:

Helmholtz first solved the problem—Given the Nabla of the velocity in any case of liquid motion, to find the velocity. His solution was that the velocity was the Nabla of the inverse square of Nabla of the Nabla of the velocity. Or, that the velocity was the inverse Nabla of the Nabla of the velocity.