Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/258

case. Therefore, in finding the differential equation between $l$ and $p,$ we may treat $b$ and ${\text{A}}'$ in (24) and $f$ and ${\text{G}}$ in (25) as constant, as well as ${\text{B}}$ and ${\text{F}}.$ These equations may be written

${\begin{aligned}4\pi ^{2}{\text{B}}{\frac {p^{2}}{l^{2}}}+bp^{2}&=4\pi ^{2}{\text{A}}',\\\pi {\text{F}}{\frac {l^{2}}{p^{2}}}+{\frac {\pi ^{2}f}{p^{2}}}&={\tfrac {1}{4}}{\text{G}}.\end{aligned}}$

Differentiating, we get

${\begin{aligned}4\pi ^{2}{\text{B}}d{\frac {p^{2}}{l^{2}}}&=-bd(p^{2}),\\\pi {\text{F}}d{\frac {l^{2}}{p^{2}}}&=-\pi ^{2}fd(p^{-2});\end{aligned}}$

or

${\begin{aligned}4\pi ^{2}{\text{B}}{\frac {p^{2}}{l^{2}}}d\log {\frac {p^{2}}{l^{2}}}&=-bp^{2}d\log p^{2},\\\pi {\text{F}}{\frac {l^{2}}{p^{2}}}d\log {\frac {l^{2}}{p^{2}}}&=-{\frac {\pi ^{2}f}{p^{2}}}dp^{-2}.\end{aligned}}$

Hence, if we write ${\text{V}}$ for the wave-velocity $(l/p)$, $n$ for the index of refraction, and $\lambda$ for the wave-length in vacuo, we have for the ratio of the two parts into which we have divided the potential energy on the elastic theory,

${\frac {bh^{2}}{4}}\div {\frac {\pi ^{2}{\text{B}}h^{2}}{l^{2}}}={\frac {d\,\log {\text{V}}}{d\,\log p}}=-{\frac {d\,\log n}{d\,\log \lambda }},$

(26)

and for the ratio of the two parts into which we have divided the kinetic energy on the electrical theory,

$\pi ^{2}{\frac {fh^{2}}{p^{2}}}\div {\frac {\pi {\text{F}}l^{2}h^{2}}{p^{2}}}={\frac {d\,\log {\text{V}}}{d\,\log p}}=-{\frac {d\,\log n}{d\,\log \lambda }}\cdot$

(27)

It is interesting to see that these ratios have the same value. This value may be expressed in another form, which is suggestive of some important relations. If we write ${\text{U}}$ for what Lord Rayleigh has called the velocity of a group of waves,^[1]

${\frac {\text{U}}{\text{V}}}=1-{\frac {d\,\log {\text{V}}}{d\,\log l}},$

${\frac {d\,\log {\text{V}}}{d\,\log l}}={\frac {{\text{V}}-{\text{U}}}{\text{V}}},$

${\frac {d\,\log {\text{V}}}{d\,\log l}}={\frac {{\text{V}}-{\text{U}}}{\text{U}}}\cdot$

(28)

It appears, therefore, that in the elastic theory that part of the potential energy which depends on the deformation expressed

1. See his "Note on Progressive Waves," Proc. Lond. Math. Soc., vol. ix, No. 125, reprinted in his Theory of Sound, vol. ii, p. 297.