We consider next the equations of motion of a continuous medium. Let be the density, the velocity components considered as functions of the co-ordinates and the time, the volume forces per unit of mass, and the stresses upon a surface perpendicular to the a-axis in the direction of increasing . Then the equations of motion are, by Newton's law, in which is the acceleration of the particle which at time has the co-ordinates . If we express this acceleration by partial differential coefficients, we obtain, after dividing by , We must show that this equation holds independently of the special choice of the Cartesian system of co-ordinates. () is a vector, and therefore is also a vector. [Pg 20] is a tensor of rank 2, is a tensor of rank 3. The second term on the left results from contraction in the indices , . The vector character of the second term on the right is obvious. In order that the first term on the right may also be a vector it is necessary for to be a tensor. Then by differentiation and contraction results, and is therefore a vector, as it also is after multiplication by the reciprocal scalar . That is a tensor, and therefore transforms according to the equation is proved in mechanics by integrating this equation over an infinitely small tetrahedron. It is also proved there, by application of the theorem of moments to an infinitely small parallelopipedon, that , and hence that the tensor of the stress is a symmetrical tensor. From what has been said it follows that, with the aid of the rules given above, the equation is co-variant with respect to orthogonal transformations in space (rotational transformations); and the rules according to which the quantities in the equation must be transformed in order that the equation may be co-variant also become evident. The co-variance of the equation of continuity, requires, from the foregoing, no particular discussion. We shall also test for co-variance the equations which express the dependence of the stress components upon the properties of [Pg 21] the matter, and set up these equations for the case of a compressible viscous fluid with the aid of the conditions of co-variance. If we neglect the viscosity, the pressure, , will be a scalar, and will depend only upon the density and the temperature of the fluid. The contribution to the stress tensor is then evidently in which is the special symmetrical tensor. This term will also be present in the case of a viscous fluid. But in this case there will also be pressure terms, which depend upon the space derivatives of the . We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be (for is a scalar). For physical reasons (no slipping) it is assumed that for symmetrical dilatations in all directions, i.e. when