Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/214

are respectively ${\tfrac {1}{2}},{\tfrac {1}{2}}$ and 0, and since at the time to be considered

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

it will appear from inspection of equations (1) that

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

(6)

This is the first part of the statical energy of the whole medium per unit of volume.

8. The second part of the statical energy of the whole medium per unit of volume (${\text{S}}_{''}$) is the space-average of $s_{''},$ which is a linear function of the twenty-seven products of $\xi ,\eta ,\zeta$ with their differential coefficients with respect to the coordinates. Now since

$\xi {\frac {d\xi }{dx}}={\tfrac {1}{2}}{\frac {d(\xi ^{2})}{dx}},$⁠$\eta {\frac {d\eta }{dx}}={\tfrac {1}{2}}{\frac {d(\eta ^{2})}{dx}},$⁠etc.,

the space-average of such products wiU be zero, and they will contribute nothing to the value of ${\text{S}}_{''}.$ There will be nine of these products, in which the same component of displacement appears twice. The remaining eighteen products may be divided into pairs according to the letters which they contain, as

$\eta {\frac {d\zeta }{dx}}$⁠and⁠$\xi {\frac {d\eta }{dx}}\cdot$

A linear function of the eighteen products may also be regarded as a linear function of the sums and differences of the products in such pairs. But since

$\eta {\frac {d\zeta }{dx}}+\zeta {\frac {d\eta }{dx}}={\frac {d(\eta \zeta )}{dx}},$

the terms of $s_{''}$ containing such sums will contribute nothing to the value of ${\text{S}}_{''}.$ We have left a linear function of the nine differences

$\eta {\frac {d\zeta }{dx}}-\zeta {\frac {d\eta }{dx}},$⁠$\zeta {\frac {d\xi }{dx}}-\xi {\frac {d\zeta }{dx}},$⁠$\xi {\frac {d\eta }{dx}}-\eta {\frac {d\xi }{dx}},$⁠etc.

(the unwritten expressions being obtained by substituting in the denominators $dy$ and $dz$ for $dx$), which constitutes the part of $s_{''}$ that we have to consider. ${\text{S}}_{''}$ is therefore a linear function of the space-averages of these nine quantities. But by (3)

$\eta {\frac {d\zeta }{dx}}-\zeta {\frac {d\eta }{dx}}={\text{L}}\left(\eta {\frac {d\zeta }{du}}-\zeta {\frac {d\eta }{du}}\right),$

and the space-average of this, at a moment of maximum displacement, is by (1)

${\frac {2\pi {\text{L}}}{l}}(\beta _{1}\gamma _{2}-\gamma _{1}\beta _{2}).$