Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/253

In other words, the displacements in either theory are subject to the same general and surface conditions as the forces required to maintain the vibrations in an element of volume in the other theory.

To fix our ideas in regard to the signification of $\Psi$ and $\Phi ,$ we may consider the case of isotropic media, in which these operators reduce to ordinary algebraic quantities, simple or complex. Now the curl of any vector necessarily satisfies the solenoidal condition (the so-called "equation of continuity"), therefore by (6) $\Psi {\mathfrak {E}}$ and ${\mathfrak {E}}$ will be solenoidal. So also will ${\mathfrak {F}}$ and $\Phi {\mathfrak {F}}$ in the electrical theory. Now for solenoidal vectors

$-{\text{curl curl }}={\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}},$

(15)

so that the equations (6) and (10) reduce to

$\Psi {\ddot {\mathfrak {E}}}=\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right){\mathfrak {E}},$

(16)

${\ddot {\mathfrak {F}}}=\left({\frac {d^{2}}{dx^{2}}}+{\frac {d^{2}}{dy^{2}}}+{\frac {d^{2}}{dz^{2}}}\right)\Psi {\mathfrak {F}}.$

(17)

For a simple train of waves, the displacement, in either theory, may be represented by a constant multiplied by

$e^{\iota (gt+ax+by+cz)}$

(18)

Our equations then reduce again to

$g^{2}\Psi {\mathfrak {E}}=(a^{2}+b^{2}+c^{2}){\mathfrak {E}},$

(19)

$g^{2}{\mathfrak {F}}=(a^{2}+b^{2}+c^{2})\Phi {\mathfrak {F}}.$

(20)

Hence

$\Psi ^{-1}=\Phi ={\frac {g^{2}}{a^{2}+b^{2}+c^{2}}}\cdot$

(21)

The last member of this equation, when real, evidently expresses the square of the velocity of light. If we set

$n^{2}=k^{2}{\frac {a^{2}+b^{2}+c^{2}}{g^{2}}},$

(22)

$k$ denoting the velocity of light in vacuo, we have

$n^{2}=k^{2}\Psi =k^{2}\Psi ^{-1}.$

(23)

When $n^{2}$ is positive, which is the case of perfectly transparent bodies, the positive root of $n^{2}$ is called the index of refraction of the medium. In the most general case, it would be appropriate to call $n$—or perhaps that root of $n^{2}$ of which the real part is positive—the (complex) index of refraction, although the terminology is hardly settled in this respect. A negative value of $n^{2}$ would represent a body from which light would be totally reflected at all angles of incidence. No such cases have been observed. Values of $n^{2}$ in which the coefficient of $\iota$ is negative, indicate media in which light is