Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/252

applied as here to a vector, the three components of the result are to be calculated separately from the three components of the operand. $-\nabla {\text{Q}}$ is therefore the electrostatic force, and $-{\text{Pot }}{\ddot {\mathfrak {F}}}$ the electrodynamic force. In establishing the equation, it was not assumed that the electrical motions are solenoidal, or such as to satisfy the so-called "equation of continuity." We may now, however, make this assumption, since it is the extreme case of the electric theory which we are to compare with the extreme case of the elastic.

It results from the definitions of curl and $\nabla$ that ${\text{curl }}\nabla {\text{Q}}=0.$ We may therefore eliminate ${\text{Q}}$ from equation (8) by taking the curl. This gives

$-{\text{curl Pot }}{\ddot {\mathfrak {F}}}=4\pi \,{\text{curl }}\Phi {\mathfrak {F}}.$

(9)

Since ${\text{curl curl}}$ and ${\frac {1}{4\pi }}{\text{Pot}}$ are inverse operators for solenoidal vectors, we may get rid of the symbol ${\text{Pot}}$ by taking the curl again. We thus get

$-{\ddot {\mathfrak {F}}}={\text{curl curl }}\Phi {\mathfrak {F}}.$

(10)

The conditions for the motion at the boundary between difiTerent media are easily obtained from the following considerations. ${\text{Pot }}{\ddot {\mathfrak {F}}}$ and ${\text{Q}}$ are evidently continuous at the interface. Therefore the components parallel to the interface of $\nabla {\text{Q}},$ and by (8) of $\Phi {\mathfrak {F}},$ will be continuous. Again, ${\text{curl Pot }}{\ddot {\mathfrak {F}}}$ is continuous at the interface, as appears from the consideration that ${\text{curl Pot }}{\dot {\mathfrak {F}}}$ is the magnetic force due to the electrical motions ${\dot {\mathfrak {F}}}.$ Therefore, by (9), ${\text{curl }}\Phi {\mathfrak {F}}$ is continuous. The solenoidal condition requires that the component of ${\mathfrak {F}}$ normal to the interface shall be continuous.

The following quantities are therefore continuous at the interface:

the components parallel to the interface of	$\Phi {\mathfrak {F}},$	$\scriptstyle {\left.{\begin{matrix}\ \\\\\ \ \end{matrix}}\right\}\,}$ (11)
the component normal to the interface of	${\mathfrak {F}},$
all components of	${\text{curl }}\Phi {\mathfrak {F}}.$

Of these conditions, the two relating to the normal components of ${\mathfrak {F}}$ and \text{curl }\Phi\mathfrak{F} are easily shown to result from the other four conditions, as in the analogous case in the elastic theory.

If we now compare in the two theories the differential equations of the motion of monochromatic light for the interior of a sensibly homogeneous medium, (6) and (10), and the special conditions for the boundary between two such media as represented by the continuity of the quantities (7) and (11), we find that these equations and conditions become identical, if

${\mathfrak {F}}=\Psi {\mathfrak {E}},$

(12)

${\mathfrak {E}}=\Phi {\mathfrak {F}},$

(13)

$\Psi =\Phi ^{-1}.$

(14)