The equations of motion of a material particle are () is a vector; , and therefore also , an invariant; thus [Pg 17] () is a vector; in the same way it may be shown that () is a vector. In general, the operation of differentiation with respect to time does not alter the tensor character. Since is an invariant (tensor of rank 0), ) is a vector, or tensor of rank 1 (by the theorem of the multiplication of tensors). If the force () has a vector character, the same holds for the difference (. These equations of motion are therefore valid in every other system of Cartesian co-ordinates in the space of reference. In the case where the forces are conservative we can easily recognize the vector character of (). For a potential energy, , exists, which depends only upon the mutual distances of the particles, and is therefore an invariant. The vector character of the force, = , is then a consequence of our general theorem about the derivative of a tensor of rank 0. Multiplying by the velocity, a tensor of rank 1, we obtain the tensor equation By contraction and multiplication by the scalar we obtain the equation of kinetic energy [Pg 18] If denotes the difference of the co-ordinates of the material particle and a point fixed in space, then the have the character of vectors. We evidently have = , so that the equations of motion of the particle may be written Multiplying this equation by we obtain a tensor equation Contracting the tensor on the left and taking the time average we obtain the virial theorem, which we shall not consider further. By interchanging the indices and subsequent subtraction, we obtain, after a simple transformation, the theorem of moments, It is evident in this way that the moment of a vector is not a vector but a tensor. On account of their skew-symmetrical character there are not nine, but only three independent equations of this system. The possibility of replacing skew-symmetrical tensors of the second rank in space of three dimensions by vectors depends upon the formation of the vector [Pg 19] If we multiply the skew-symmetrical tensor of rank 2 by the special skew-symmetrical tensor introduced above, and contract twice, a vector results whose components are numerically equal to those of the tensor. These are the so-called axial vectors which transform differently, from a right-handed system to a left-handed system, from the . There is a gain in picturesqueness in regarding a skew-symmetrical tensor of rank 2 as a vector in space of three dimensions, but it does not represent the exact nature of the corresponding quantity so well as considering it a tensor.