Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/201

easily appear that the manner in which the general or average flux in any element ${\text{D}}v$ (represented by $\xi ,\eta ,\zeta$) distributes itself among the molecules and intermolecular spaces must be entirely determined by the amount and direction of that flux and its period of oscillation. Hence, and on account of the superposable character of the motions which we are considering, we may conclude that the values of $\xi ',\eta ',\zeta '$ at any given point in the medium are capable of expression as linear functions of $\xi ,\eta ,\zeta$ in a manner which shall be independent of the time and of the orientation of the wave-planes and the distance of a nodal plane from the point considered, so long as the period, of oscillation remains the same. But a change in the period may presumably affect the relation between $\xi ',\eta ',\zeta '$ and $\xi ,\eta ,\zeta$ to a certain extent. And the relation between $\xi ',\eta ',\zeta '$ and $\xi ,\eta ,\zeta$ will vary rapidly as we pass from one point to another within the element ${\text{D}}v.$

5. In the motion which we are considering there occur alternately instants of no velocity and instants of no displacement. The statical energy of the medium at an instant of no velocity must be equal to its kinetic energy at an instant of no displacement. Let us examine each of these quantities, and consider the equation which expresses their equality.

6. Since in every part of an element ${\text{D}}v$ the irregular as well as the regular part of the displacement is entirely determined (for light of a given period) by the values of $\xi ,\eta ,\zeta ,$ the statical energy of the element must be a quadratic function of $\xi ,\eta ,\zeta ,$ say

$({\text{A}}\xi ^{2}+{\text{B}}\eta ^{2}+{\text{C}}\zeta ^{2}+{\text{E}}\eta \zeta +{\text{F}}\zeta \xi +{\text{G}}\xi \zeta ){\text{D}}v,$

where ${\text{A, B, }}$ etc. depend only on the nature of the medium and the period of oscillation. At an instant of no velocity, when

$\sin 2\pi {\frac {t}{p}}=0,$and$\cos ^{2}2\pi {\frac {t}{p}}=1,$

the above expression will reduce by equations (1) to

$({\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta )\cos ^{2}2\pi {\frac {u}{l}}{\text{D}}v.$

Since the average value of $\cos ^{2}2\pi {\frac {u}{l}}$ in an indefinitely extended space is ${\tfrac {1}{2}},$ we have for the statical energy in a unit of volume

${\text{S}}={\tfrac {1}{2}}({\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta ).$

(2)

7. The kinetic energy of the whole medium is represented by the double volume-integral^[1]

${\tfrac {1}{2}}\textstyle \sum \displaystyle \iint {\frac {({\dot {\xi }}+{\dot {\xi '}})_{1}({\dot {\xi }}+{\dot {\xi '}})_{2}}{r}}dv_{1}dv_{2},$

1. The fluxes are supposed to be measured by the electromagnetic system of units. It is to be observed that the difference of opinion which has prevailed with respect to the estimation of the energy of electrical currents does not extend to such as are solenoidal, which may be regarded as composed of closed circuits.