Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/231

density. For the average values of these components in the small spaces defined above, we may write

$-{\text{Pot }}{\ddot {\xi }}_{\text{Ave}},$⁠$-{\text{Pot }}{\ddot {\eta }}_{\text{Ave}},$⁠$-{\text{Pot }}{\ddot {\zeta }}_{\text{Ave}},$

since it will make no difference whether we take the average before or after the operation of taking the potential.

6. If we write ${\text{X, Y, Z}}$ for the components of the total electromotive force (electrostatic and electrodynamic), we have

$[{\text{X}}]_{\text{Ave}}=-{\text{Pot }}[{\ddot {\xi }}]_{\text{Ave}}-{\frac {d[q]_{\text{Ave}}}{dx}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (3)
etc.,

or by (2)

$[{\text{X}}]_{\text{Ave}}={\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}[{\xi }]_{\text{Ave}}-{\frac {d[q]_{\text{Ave}}}{dx}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (4)
etc.,

It will be convenient to represent these relations by a vector notation. If we represent the displacement by ${\mathsf {U}},$ and the electromotive force by ${\mathsf {E}},$ the three equations of (3) will be represented by the single vector equation

$[{\mathsf {E}}]_{\text{Ave}}=-{\text{Pot }}[{\ddot {\mathsf {U}}}]_{\text{Ave}}-\nabla [q]_{\text{Ave}},$

(5)

and the three equations of (4) by the single vector equation

$[{\mathsf {E}}]_{\text{Ave}}={\frac {4\pi ^{2}}{p^{2}}}{\text{Pot }}[{\mathsf {U}}]_{\text{Ave}}-\nabla [q]_{\text{Ave}},$

(5)

where, in accordance with quatemionic usage, $\nabla [q]_{\text{Ave}}$ represents the vector which has for components the derivatives of $[q]_{\text{Ave}}$ with respect to rectangular coordinatea The symbol $Pot$ in such a vector equation signifies that the operation which is denoted by this symbol in a scalar equation is to be performed upon each of the components of the vector. 7. We may here observe that if we are not satisfied with the law adopted for the determination of electrodynamic force we have only to substitute for $-Pot$ in these vector equations, and in those which follow, the symbol for the operation, whatever it may be, by which we calculate the electrodynamic force from the acceleration.^[1] For the operation must be of such a character that if the acceleration consist of any number of parts, the force due to the whole acceleration will be the resultant of the forces due to the separate parts. It will evidently make no difference whether we take an average before or after such an operation.

1. The same would not be true of the corresponding scalar equations, (3) and (4). For one component of the force might depend upon all the components of acceleration. Such is in fact the case with the law of electromotive force proposed by Weber.