Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/207

infinitesimal variation in the type of a vibration, due to a constraint, will not affect the period. If we first consider a certain system of stationary waves, then a system in which the wave-length is greater by an infinitesimal $dl$ (the direction of oscillation remaining the same), the period will be increased by an infinitesimal $dp,$ and the manner in which the flux distributes itself among the molecules and intermolecular spaces will presumably be infinitesimally changed. But if we suppose that in the second system of waves there is applied a constraint compelling the flux to distribute itself in the same way among the molecules and intermolecular spaces as in the first system (so that $\xi ',\eta ',\zeta '$ shall be the same functions as before of $\xi ,\eta ,\zeta ,$—a supposition perfectly compatible with the fact that the values of $\xi ,\eta ,\zeta$ are changed), this constraint, according to the principle cited, will not affect the period of oscillation. Our equations will apply to such a constrained type of oscillation, and ${\text{A, B, }}$ etc., and ${\text{A}}',{\text{B}}',$ etc., and therefore ${\text{H}}$ and ${\text{H}}',$ will have the same values in the last described system of waves as in the first system, although the wave-length and the period have been varied. Therefore, in differentiating equation (18), which is essentially an equation between $l$ and $p,$ or its equivalent (19), we may treat ${\text{H}}$ and ${\text{H}}',$ as constant. This gives

$-{\frac {2}{n^{3}}}{\frac {dn}{d\lambda }}={\frac {4\pi {\text{H}}'}{\lambda ^{3}}}\cdot$

We thus obtain the values of ${\text{H}}$ and ${\text{H}}'$

${\text{H}}'=-{\frac {\lambda ^{3}}{2\pi n^{3}}}{\frac {dn}{d\lambda }},$⁠${\text{H}}={\frac {2\pi k^{2}}{n^{2}}}-{\frac {2\pi k^{3}\lambda }{n^{3}}}{\frac {dn}{d\lambda }}\cdot$

(20)

By determining the values of ${\text{H}}$ and ${\text{H}}'$ for diflerent directions of oscillation, we may determine the values of ${\text{A, B, }}$ etc., and ${\text{A}}',{\text{B}}',$ etc.

By means of these equations, the ratios of the statical energy (${\text{S}}$), the kinetic energy due to the regular part of the flux (${\text{T}}$), and the kinetic energy due to the irregular part of the flux (${\text{T}}'$), are easily obtained in a form which admits of experimental determination. EquationS (8) and (9) give

${\text{T}}={\frac {\pi \rho ^{2}l^{2}}{p^{2}}},$⁠${\text{T}}'={\frac {2\pi ^{2}{\text{H}}'\rho ^{2}}{p^{2}}}\cdot$

Therefore, by (20),

${\frac {{\text{T}}'}{\text{T}}}={\frac {2\pi {\text{H}}'}{l^{2}}}={\frac {2\pi {\text{H}}'n^{2}}{\lambda ^{2}}}=-{\frac {\lambda }{n}}{\frac {dn}{d\lambda }}=-{\frac {d\log n}{d\log \lambda }}\cdot$

(21)

${\frac {\text{S}}{\text{T}}}={\frac {{\text{T}}+{\text{T}}'}{\text{T}}}=1+{\frac {{\text{T}}'}{\text{T}}}={\frac {d\log \lambda -d\log n}{d\log \lambda }}={\frac {d\log l}{d\log \lambda }}\cdot$

(22)

${\frac {{\text{T}}'}{\text{S}}}=-{\frac {d\log n}{d\log l}}\cdot$

(23)