Analytic tensors of higher rank (number of indices) may be defined. It is possible and advantageous to regard vectors as tensors of rank 1, and invariants (scalars) as tensors of rank 0. In this respect, the problem of the theory of invariants may be so formulated: according to what laws may new tensors be formed from given tensors? We shall consider these laws now, in order to be able to apply them later. We shall deal first only with the properties of tensors with respect to the transformation from one Cartesian system to another in the same space of reference, by means of linear orthogonal transformations. As the laws are wholly independent of the number of dimensions, we shall leave this number, , indefinite at first. Definition. If a figure is defined with respect to every system of Cartesian co-ordinates in a space of reference of dimensions by the numbers ( = number of indices), then these numbers are the components of a tensor of rank if the transformation law is [Pg 13] Remark. From this definition it follows that is an invariant, provided that (), (), () ... are vectors. Conversely, the tensor character of () may be inferred, if it is known that the expression (8) leads to an invariant for an arbitrary choice of the vectors (), (), etc. Addition and Subtraction. By addition and subtraction of the corresponding components of tensors of the same rank, a tensor of equal rank results: The proof follows from the definition of a tensor given above. Multiplication. From a tensor of rank and a tensor of rank we may obtain a tensor of rank + by multiplying all the components of the first tensor by all the components of the second tensor: Contraction. A tensor of rank - 2 may be obtained from one of rank by putting two definite indices equal to each other and then summing for this single index: [Pg 14] The proof is In addition to these elementary rules of operation there is also the formation of tensors by differentiation ("erweiterung"): New tensors, in respect to linear orthogonal transformations, may be formed from tensors according to these rules of operation. Symmetrical Properties of Tensors. Tensors are called symmetrical or skew-symmetrical in respect to two of their indices, and , if both the components which result from interchanging the indices and are equal to each other or equal with opposite signs. Theorem. The character of symmetry or skew-symmetry exists independently of the choice of co-ordinates, and in this lies its importance. The proof follows from the equation defining tensors. Special Tensors. I. The quantities (4) are tensor components (fundamental tensor). [Pg 15] Proof. If in the right-hand side of the equation of transformation = , we substitute for the quantities (which are equal to 1 or 0 according as = or ≠ ), we get