Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/204

indicates that the we may regard the infinitesimal element $dv$ the energy (due to this part of the flux)

${\tfrac {1}{2}}\textstyle \sum \displaystyle {\dot {\xi '}}{\text{ Pot }}{\dot {\xi '}}dv.$

Let us consider the energy due to the irregular flux which will belong to the above defined element ${\text{D}}v,$ which is not infinitely small, but which has the advantage of being one of physically similar elements which make up the whole medium. The energy of this element is found by adding the energies of all the infinitesimal elements of which it is composed Since these are quadratic functions of the quantities ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }},$ which are sensibly constant throughout the element ${\text{D}}v,$ the sum will be a quadratic function of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }},$ say

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

which will therefore represent the energy of the element ${\text{D}}v$ due to the irregular flux. The coefficients ${\text{A}}',{\text{B}}',$ etc., are determined by the nature of the medium and the period of oscillation. They will be constant throughout the medium, since one element ${\text{D}}v$ does not differ from another.

This expression reduces by equations (4) to

${\frac {4\pi ^{2}}{p^{2}}}({\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta )\cos ^{2}2\pi {\frac {u}{l}}{\text{D}}v.$

The kinetic energy of the irregular flux in a unit of volume is therefore

${\text{T}}'={\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta ).$

(9)

10. Equating the statical and kinetic energies, we have

${\tfrac {1}{2}}({\text{A}}\alpha ^{2}+{\text{B}}\beta ^{2}+{\text{C}}\gamma ^{2}+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta )$ $={\frac {\pi l^{2}}{p^{2}}}(\alpha ^{2}+\beta ^{2}+\gamma ^{2})+{\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta ).$

(10)

The velocity (${\text{V}}$) of the corresponding system of progressive waves is given by the equation

${\begin{aligned}{\text{V}}^{2}={\frac {l^{2}}{p^{2}}}&={\frac {1}{2\pi }}{\frac {+{\text{E}}\beta \gamma +{\text{F}}\gamma \alpha +{\text{G}}\alpha \beta }{\alpha ^{2}+\beta ^{2}+\gamma ^{2}}}\\&-{\frac {2\pi }{p^{2}}}{\frac {{\text{A}}'\alpha ^{2}+{\text{B}}'\beta ^{2}+{\text{C}}'\gamma ^{2}+{\text{E}}'\beta \gamma +{\text{F}}'\gamma \alpha +{\text{G}}'\alpha \beta }{\alpha ^{2}+\beta ^{2}+\gamma ^{2}}}\cdot \end{aligned}}$

(11)

If we set

$a={\frac {1}{2\pi }}{\text{A}}-{\frac {2\pi }{p^{2}}}{\text{A}}',$⁠$b={\frac {1}{2\pi }}{\text{B}}-{\frac {2\pi }{p^{2}}}{\text{B}}',$⁠etc.,

(12)

and

$\rho ^{2}=\alpha ^{2}+\beta ^{2}+\gamma ^{2},$

the equation reduces to

${\text{V}}^{2}={\frac {a\alpha ^{2}+b\beta ^{2}+c\gamma ^{2}+e\beta \gamma +f\gamma \alpha +g\alpha \beta }{\rho ^{2}}}\cdot$

(13)