That there are, nevertheless, in the general case, invariant differential operations for tensors, is recognized most satisfactorily in the following way, introduced by Levi-Civita and Weyl. Let () be a contra-variant vector whose components are given with respect to the co-ordinate system of the . Let and [Pg 73] be two infinitesimally near points of the continuum. For the infinitesimal region surrounding the point , there is, according to our way of considering the matter, a co-ordinate system of the (with imaginary -co-ordinate) for which the continuum is Euclidean. Let be the co-ordinates of the vector at the point . Imagine a vector drawn at the point , using the local system of the , with the same co-ordinates (parallel vector through ), then this parallel vector is uniquely determined by the vector at and the displacement. We designate this operation, whose uniqueness will appear in the sequel, the parallel displacement of the vector from to the infinitesimally near point . If we form the vector difference of the vector () at the point and the vector obtained by parallel displacement from to , we get a vector which may be regarded as the differential of the vector () for the given displacement . This vector displacement can naturally also be considered with respect to the co-ordinate system of the . If are the co-ordinates of the vector at , the co-ordinates of the vector displaced to along the interval (), then the do not vanish in this case. We know of these quantities, which do not have a vector character, that they must depend linearly and homogeneously upon the and the . We therefore put In addition, we can state that the must be symmetrical with respect to the indices and . For we can assume from a representation by the aid of a Euclidean system of local co-ordinates that the same parallelogram will be described by the displacement of an element along a second element [Pg 74] as by a displacement of along . We must therefore have The statement made above follows from this, after interchanging the indices of summation, and , on the right-hand side. Since the quantities determine all the metrical properties of the continuum, they must also determine the . If we consider the invariant of the vector that is, the square of its magnitude, which is an invariant, this cannot change in a parallel displacement. We therefore have or, by (67), Owing to the symmetry of the expression in the brackets with respect to the indices and , this equation can be valid for an arbitrary choice of the vectors () and only when the expression in the brackets vanishes for all combinations of the indices. By a cyclic interchange of the indices , , , we obtain thus altogether three equations, from which we obtain, on taking into account the symmetrical property of the , [Pg 75] in which, following Christoffel, the abbreviation has been used, If we multiply (68) by and sum over the , we obtain