According to what has been said, it is evident that the formulation of the general theory of relativity assumes a generalization of the theory of invariants and the theory of tensors; the question is raised as to the form of the equations which are co-variant with respect to arbitrary point transformations. The generalized calculus of tensors was developed by mathematicians long before the theory of relativity. Riemann first extended Gauss's train of thought to continua of any number of dimensions; with prophetic vision he saw the physical meaning of this generalization of Euclid's geometry. Then followed the development of the theory in the form of the calculus of tensors, particularly by Ricci and Levi-Civita. This is the place for a brief presentation of the most important mathematical concepts and operations of this calculus of tensors. We designate four quantities, which are defined as functions of the with respect to every system of co-ordinates, as components, [Pg 68] , of a contra-variant vector, if they transform in a change of co-ordinates as the co-ordinate differentials . We therefore have Besides these contra-variant vectors, there are also co-variant vectors. If are the components of a co-variant vector, these vectors are transformed according to the rule The definition of a co-variant vector is chosen in such a way that a co-variant vector and a contra-variant vector together form a scalar according to the scheme, Accordingly, In particular, the derivatives of a scalar , are components of a co-variant vector, which, with the co-ordinate differentials, form the scalar ; we see from this example how natural is the definition of the co-variant vectors. There are here, also, tensors of any rank, which may have co-variant or contra-variant character with respect to each index; as with vectors, the character is designated by the position [Pg 69] of the index. For example, denotes a tensor of the second rank, which is co-variant with respect to the index , and contra-variant with respect to the index . The tensor character indicates that the equation of transformation is Tensors may be formed by the addition and subtraction of tensors of equal rank and like character, as in the theory of invariants of orthogonal linear substitutions, for example, The proof of the tensor character of depends upon (58). Tensors may be formed by multiplication, keeping the character of the indices, just as in the theory of invariants of linear orthogonal transformations, for example, The proof follows directly from the rule of transformation. Tensors may be formed by contraction with respect to two indices of different character, for example, The tensor character of determines the tensor character of . Proof— [Pg 70] The properties of symmetry and skew-symmetry of a tensor with respect to two indices of like character have the same significance as in the theory of invariants.