Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/213

we may regard $\sigma$ as a quadratic function of $\xi ,\eta ,\zeta$ and diff, coeff., or as a linear function of the seventy-eight squares and products of these quantities. But the seventy-eight coefficients by which this function is expressed will vary with the position of the element of volume with respect to the surrounding molecules.

In estimating the statical energy for any considerable space by the integral

$\int \sigma dv,$

it will be allowable to substitute for the seventy-eight coefficients contained implicitly in $\sigma$ their average values throughout the medium. That is, if we write $s$ for a quadratic function of $\xi ,\eta ,\zeta$ and diff, coeff. in which the seventy-eight coefficients are the space-averages of those in $\sigma ,$ the statical energy of any considerable space may be estimated by the integral

$\int sdv.$

(This will appear most distinctly if we suppose the integration to be first effected for a thin slice of the medium bounded by two wave-planes.) The seventy-eight coefficients of this function $s$ are determined solely by the nature of the medium and the period of oscillation.

We may divide $s$ into three parts, of which the first ($s_{'}$) contains the squares and products of $\xi ,\eta ,\zeta ,$ the second ($s_{''}$) contains the products of $\xi ,\eta ,\zeta$ with the differential coefficients, and the third ($s_{'''}$) contains the squares and products of the differential coefficients. It is evident that the average statical energy of the whole medium per unit of volume is the space-average of $s,$ and that it will consist of three parts, which are the space-averages of $s_{'},s_{''},$ and $s_{'''},$ respectively. These parts we may call ${\text{S}}_{'},{\text{S}}_{''},$ and ${\text{S}}_{'''}.$ Only the first of these was considered in the preceding paper.

Now the considerations which justify us in neglecting, for an approximate estimate, the terms of $s$ which contain the differential coefficients of $\xi ,\eta ,\zeta$ with respect to the coordinates, will apply with especial force to the terms which contain the squares and products of these differential coefficients. Therefore, to carry the approximation one step beyond that of the preceding paper, it will only be necessary to take account of $s_{'}$ and $s_{''}$ and of ${\text{S}}_{'}$ and ${\text{S}}_{''}.$

7. We may set

$s_{'}={\text{A}}\xi ^{2}+{\text{B}}\eta ^{2}+{\text{C}}\zeta ^{2}+{\text{E}}\eta \zeta +{\text{F}}\zeta \xi +{\text{G}}\xi \eta ,$

(5)

where, for a given medium and light of a given period, ${\text{A, B, C, D, E, F, G}}$ are constant.

Since the average values of

$\sin ^{2}2\pi {\frac {u}{l}},\,\cos ^{2}2\pi {\frac {u}{l}},\,\,\sin 2\pi {\frac {u}{l}}\cos 2\pi {\frac {u}{l}}$