By the introduction of the imaginary time variable, , Minkowski has made the theory of invariants for the four-dimensional continuum of physical phenomena fully analogous to the theory of invariants for the three-dimensional continuum of Euclidean space. The theory of four-dimensional tensors of special relativity differs from the theory of tensors in three-dimensional space, therefore, only in the number of dimensions and the relations of reality. A physical entity which is specified by four quantities, , in an arbitrary inertial system of the , , , , is called a 4-vector, with the components , if the correspond in their relations of reality and the properties of transformation to the ; it may be of the nature of a space or of a time. The sixteen quantities , then form the components of a tensor of the second rank, if they transform according to the scheme It follows from this that the behave, with respect to their properties of transformation and their properties of reality, as the products of components, of two 4-vectors, () and (). All the components are real except those which contain the index 4 once, those being purely imaginary. Tensors [Pg 41] of the third and higher ranks may be defined in an analogous way. The operations of addition, subtraction, multiplication, contraction and differentiation for these tensors are wholly analogous to the corresponding operations for tensors in three-dimensional space. Before we apply the tensor theory to the four-dimensional space-time continuum, we shall examine more particularly the skew-symmetrical tensors. The tensor of the second rank has, in general, 16 = 4·4 components. In the case of skew-symmetry the components with two equal indices vanish, and the components with unequal indices are equal and opposite in pairs. There exist, therefore, only six independent components, as is the case in the electromagnetic field. In fact, it will be shown when we consider Maxwell's equations that these may be looked upon as tensor equations, provided we regard the electromagnetic field as a skew-symmetrical tensor. Further, it is clear that the skew-symmetrical tensor of the third rank (skew-symmetrical in all pairs of indices) has only four independent components, since there are only four combinations of three different indices. We now turn to Maxwell's equations (19a), (19b), (20a), (20b), and introduce the notation:[10] with the convention that shall be equal to . Then [Pg 42] Maxwell's equations may be combined into the forms