must hold. We shall now show that this equation leads to the same law of motion of a material particle as that already obtained. Let us imagine the matter to be of infinitely small extent in space, that is, a four-dimensional thread; then by integration over the whole thread with respect to the space co-ordinates , , , we obtain Now is an invariant, as is, therefore, also . We shall calculate this integral, first with respect to the inertial system which we have chosen, and second, with respect to a system relatively to which the matter has the velocity zero. The integration is to be extended over a filament [Pg 56] of the thread for which may be regarded as constant over the whole section. If the space volumes of the filament referred to the two systems are and respectively, then we have and therefore also If we substitute the right-hand side for the left-hand side in the former integral, and put outside the sign of integration, we obtain, We see, therefore, that the generalized conception of the energy tensor is in agreement with our former result. The Eulerian Equations for Perfect Fluids. In order to get nearer to the behaviour of real matter we must add to the energy tensor a term which corresponds to the pressures. The simplest case is that of a perfect fluid in which the pressure is determined by a scalar . Since the tangential stresses , etc., vanish in this case, the contribution to the energy tensor must be of the form . We must therefore put [Pg 57] At rest, the density of the matter, or the energy per unit volume, is in this case, not but . For In the absence of any force, we have If we multiply this equation by and sum for the 's we obtain, using (40), where we have put . This is the equation of continuity, which differs from that of classical mechanics by the term , which, practically, is vanishingly small. Observing (52), the conservation principles take the form 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. [Pg 58] LECTURE III THE GENERAL THEORY OF RELATIVITY