With this, everything essential has been said with regard to the algebraic properties of tensors. The Fundamental Tensor. It follows from the invariance of for an arbitrary choice of the , in connexion with the condition of symmetry consistent with (55), that the are components of a symmetrical co-variant tensor (Fundamental Tensor). Let us form the determinant, , of the , and also the minors, divided by , corresponding to the single . These minors, divided by , will be denoted by , and their co-variant character is not yet known. Then we have If we form the infinitely small quantities (co-variant vectors) multiply by and sum over the , we obtain, by the use of (62), Since the ratios of the , are arbitrary, and the as well as the are components of vectors, it follows that the are the components of a contra-variant tensor[15] (contra-variant fundamental tensor). The tensor character of (mixed fundamental [Pg 71] tensor) accordingly follows, by (62). By means of the fundamental tensor, instead of tensors with co-variant index character, we can introduce tensors with contra-variant index character, and conversely. For example, [15]If we multiply (64) by , sum over the , and replace the by a transformation to the accented system, we obtain The statement made above follows from this, since, by (64), we must also have and both equations must hold for every choice of . Volume Invariants. The volume element is not an invariant. For by Jacobi's theorem, But we can complement so that it becomes an invariant. If we form the determinant of the quantities [Pg 72] we obtain, by a double application of the theorem of multiplication of determinants, We therefore get the invariant, Formation of Tensors by Differentiation. Although the algebraic operations of tensor formation have proved to be as simple as in the special case of invariance with respect to linear orthogonal transformations, nevertheless in the general case, the invariant differential operations are, unfortunately, considerably more complicated. The reason for this is as follows. If is a contra-variant vector, the coefficients of its transformation, , are independent of position only if the transformation is a linear one. For then the vector components, , at a neighbouring point transform in the same way as the , from which follows the vector character of the vector differentials, and the tensor character of . But if the are variable this is no longer true.