Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/217

Yet, as in the case of the statical energy, we may substitute the average values of these coefficients for the coefficients themselves in the integral by which we obtain the energy of any considerable space. The kinetic energy due to the irregular part of the flux is thus reduced to a quadratic function of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ and diff. coeff. which has constant coefficients for a given medium and light of a given period.

The function may be divided into three parts, of which the first contains the squares and products of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }},$ the second the products of ${\dot {\xi }},{\dot {\eta }},{\dot {\zeta }}$ with their differential coefficients, and the third, which may be neglected, the squares and products of the differential coefficients.

We may proceed with the reduction precisely as in the case of the statical energy, except that the differentiations with respect to the time will introduce the constant factor ${\frac {4\pi ^{2}}{p^{2}}}\cdot$ This will give for the first part of the kinetic energy of the irregular flux per unit of volume

${\text{T}}'={\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha _{1}^{2}+{\text{B}}'\beta _{1}^{2}+{\text{C}}'\gamma _{1}^{2}+{\text{E}}'\beta _{1}\gamma _{1}+{\text{F}}'\gamma _{1}\alpha _{1}+{\text{G}}'\alpha _{1}\beta _{1})$ $+{\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha _{2}^{2}+{\text{B}}'\beta _{2}^{2}+{\text{C}}'\gamma _{2}^{2}+{\text{E}}'\beta _{2}\gamma _{2}+{\text{F}}'\gamma _{2}\alpha _{2}+{\text{G}}'\alpha _{2}\beta _{2}),$

(11)

and for the second part of the same

${\begin{aligned}{\text{T}}_{''}'&={\frac {4\pi ^{2}\Phi '}{p^{2}l}}\Theta \\&={\frac {4\pi ^{2}\Phi '}{p^{2}l}}[{\text{L}}(\beta _{1}\gamma _{2}-\gamma _{1}\beta _{2})+{\text{M}}(\gamma _{1}\alpha _{2}-\alpha _{1}\gamma _{2})+{\text{N}}(\alpha _{1}\beta _{2}-\beta _{1}\alpha _{2})],\end{aligned}}$

(12)

where ${\text{A}}',{\text{B}}',{\text{C}}',{\text{E}}',{\text{F}}',{\text{G}}'$ are constant, and $\Phi '$ a quadratic function of ${\text{L, M,}}$ and ${\text{N}},$ for a given medium and light of a given period.

12. Equating the statical and kinetic energies, we have

${\text{S}}_{'}+{\text{S}}_{''}={\text{T}}+{\text{T}}_{'}'+{\text{T}}_{''}',$

that is, by equations (6), (9), (10), (11), and (12),

${\begin{aligned}&{\tfrac {1}{2}}({\text{A}}\alpha _{1}^{2}+{\text{B}}\beta _{1}^{2}+{\text{C}}\gamma _{1}^{2}+{\text{E}}\beta _{1}\gamma _{1}+{\text{F}}\gamma _{1}\alpha _{1}+{\text{G}}\alpha _{1}\beta _{1})\\+&{\tfrac {1}{2}}({\text{A}}\alpha _{2}^{2}+{\text{B}}\beta _{2}^{2}+{\text{C}}\gamma _{2}^{2}+{\text{E}}\beta _{2}\gamma _{2}+{\text{F}}\gamma _{2}\alpha _{2}+{\text{G}}\alpha _{2}\beta _{2})\\=&{\frac {\Phi }{l}}[{\text{L}}(\beta _{1}\gamma _{2}-\gamma _{1}\beta _{2})+{\text{M}}(\gamma _{1}\alpha _{2}-\alpha _{1}\gamma _{2})+{\text{N}}(\alpha _{1}\beta _{2}-\beta _{1}\alpha _{2})]\\=&{\frac {\pi l^{2}}{p^{2}}}(\alpha _{1}^{2}+\beta _{1}^{2}+\gamma _{1}^{2}+\alpha _{2}^{2}+\beta _{2}^{2}+\gamma _{2}^{2})\\+&{\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha _{1}^{2}+{\text{B}}'\beta _{1}^{2}+{\text{C}}'\gamma _{1}^{2}+{\text{E}}'\beta _{1}\gamma _{1}+{\text{F}}'\gamma _{1}\alpha _{1}+{\text{G}}'\alpha _{1}\beta _{1})\\+&{\frac {2\pi ^{2}}{p^{2}}}({\text{A}}'\alpha _{2}^{2}+{\text{B}}'\beta _{2}^{2}+{\text{C}}'\gamma _{2}^{2}+{\text{E}}'\beta _{2}\gamma _{2}+{\text{F}}'\gamma _{2}\alpha _{2}+{\text{G}}'\alpha _{2}\beta _{2})\\+&{\frac {4\pi ^{2}\Phi '}{p^{2}l}}[{\text{L}}(\beta _{1}\gamma _{2}-\gamma _{1}\beta _{2})+{\text{M}}(\gamma _{1}\alpha _{2}-\alpha _{1}\gamma _{2})+{\text{N}}(\alpha _{1}\beta _{2}-\beta _{1}\alpha _{2})]\end{aligned}}$

(13)