Page:ComstockInertia.djvu/4

This can be readily shown by a consideration of the figure. When the velocity of the element is along (x) there is an amount of work $(v_{x}\cdot X_{x})dy\cdot dz$ done per second on the element by the tension ($X_{x}$) applied at the surface ($O'A'$), and this energy is instantly available at the surface ($OA$), where it is given out. The distance over which the energy is transmitted being $dx$ (the thickness of the element), the rate of energy-flow is

$-v_{x}X_{x}dydzdx=-v_{x}X_{x}d\tau \,$,

where ($d\tau$) is the element of volume.

In like manner the velocity ($v_{y}$) and the shearing stress ($Y_{x}\cdot dy\cdot dz$) cause energy to be taken up at the surface ($O'A'$) and given out at the surface ($OA$), and we have the rate of flow along the x-axis

$-v_{y}Y_{x}d\tau \,$;

and finally the velocity ($v_{z}$) and the shearing stress ($Z_{x}\cdot dy\cdot dz$) give

$-v_{z}Z_{x}d\tau \,$.

Hence adding we have, if we call ($f'_{x}$) the density of flow along $x$,

$f'_{x}d\tau =-(v_{x}X_{x}+v_{y}Y_{x}+v_{z}Z_{x})d\tau \,$.

Obtaining the corresponding equations in similar way we have finally for the three components of the density of energyflow along the constraints in any system

${\begin{cases}f'_{x}=-(v_{x}X_{x}+v_{y}Y_{x}+v_{z}Z_{x}),\\f'_{y}=-(v_{x}X_{y}+v_{y}Y_{y}+v_{z}Z_{y}),\\f'_{z}=-(v_{x}X_{z}+v_{y}Y_{z}+v_{z}Z_{z}),\end{cases}}$

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