Page:The Meaning of Relativity - Albert Einstein (1922).djvu/72

where we have put ${\frac {\delta p}{\delta x_{\mu }}}{\frac {dx_{\mu }}{d\tau }}={\frac {dp}{d\tau }}$. This is the equation of continuity, which differs from that of classical mechanics by the term ${\frac {dp}{d\tau }}$, which, practically, is vanishingly small. Observing (52), the conservation principles take the form

$\sigma {\frac {du_{\mu }}{d\tau }}+u_{\mu }{\frac {dp}{d\tau }}+{\frac {\delta p}{\delta x_{\mu }}}=0.$

(53)

The equations for the first three indices evidently correspond to the Eulerian equations. That the equations (52) and (53) correspond, to a first approximation, to the hydrodynamical equations of classical mechanics, is a further confirmation of the generalized energy principle. The density of matter and of energy has the character of a symmetrical tensor.