We thus arrive at the law of gravitation for continuous matter~\Eq{(46.6)} \index{Matter!identification of}% but with a different justification. Appeal is now made to a Principle of Identification. Our deductive theory starts with the interval (introduced by the fundamental axiom of \SecRef{1}), from which the tensor~$g_{\mu\nu}$ is immediately obtained. By pure mathematics we derive other tensors $G_{\mu\nu}$, $B_{\mu\nu\sigma\rho}$, and if necessary more complicated tensors. These constitute our world-building material; and the aim of the deductive theory is to construct from this a world which functions in the same way as the known physical world. If we succeed, mass, momentum, stress, etc.\ must be the vulgar names for certain analytical quantities in the deductive theory; and it is this stage of naming the analytical tensors which is reached in~\Eq{(54.3)}. If the theory provides a tensor $G_{\mu}^{\nu} - \frac{1}{2} g_{\mu}^{\nu} G$ which behaves in exactly the same way as the tensor \PageSep{120} summarising the mass, momentum and stress of matter is observed to behave, it is difficult to see how anything more could be required of it\footnotemark.\footnotetext {For a complete theory it would be necessary to show that matter as now defined has a \index{Atomicity@Atomicity|indexfn}% tendency to aggregate into atoms leaving large tracts of the world vacant. The relativity theory has not yet succeeded in finding any clue to the phenomenon of atomicity.} By means of \Eq{(53.91)} and~\Eq{(54.3)} the physical quantities $\rho$, $u$, $v$, $w$, $p_{xx}$\Add{,}~\dots $p_{zz}$ are identified in terms of the fundamental tensors of space-time. There are $10$~of these physical quantities and $10$~different components of $G_{\mu}^{\nu} - \frac{1}{2}g_{\mu}^{\nu} G$, so that the identification is just sufficient. It will be noticed that this identification gives a dynamical, not a kinematical definition of the velocity of matter $u$, $v$,~$w$; it is appropriate, for example, to the case of a rotating homogeneous and continuous fly-wheel, in which there is no velocity of matter in the kinematical sense, although a dynamical velocity is indicated by its gyrostatic \index{Dynamical velocity}% properties\footnotemark.\footnotetext {\Title{Space, Time and Gravitation}, p.~194.} The connection with the ordinary kinematical velocity, which \index{Kinematical velocity}% determines the direction of the world-line of a particle in four dimensions, is followed out in \SecRef{56}. Contracting \Eq{(54.3)} by setting $\nu = \mu$, and remembering that $g_{\mu}^{\mu} = 4$, we have \[ G = 8\pi T, \Tag{(54.4)} \] so that an equivalent form of~\Eq{(54.3)} is \[ G_{\mu}^{\nu} = -8\pi(T_{\mu}^{\nu} - \tfrac{1}{2} g_{\mu}^{\nu}T). \Tag{(54.5)} \] When there is no material energy-tensor this gives \[ G_{\mu}^{\nu} = 0, \] which is equivalent to Einstein's law $G_{\mu\nu} = 0$ for empty space.