The justification for the last sign of equality becomes evident if one applies (4) to the inverse substitution (5). II. There is a tensor () skew-symmetrical with respect to all pairs of indices, whose rank is equal to the number of dimensions, , and whose components are equal to +1 or -1 according as is an even or odd permutation of 1 2 3.... The proof follows with the aid of the theorem proved above These few simple theorems form the apparatus from the theory of invariants for building the equations of pre-relativity physics and the theory of special relativity. We have seen that in pre-relativity physics, in order to specify relations in space, a body of reference, or a space of reference, is required, and, in addition, a Cartesian system of co-ordinates. We can fuse both these concepts into a single one by thinking of a Cartesian system of co-ordinates as a cubical frame-work formed of rods each of unit length. The co-ordinates of the lattice points of this frame are integral numbers. It follows from the fundamental relation that the members of such a space-lattice are all of unit length. To specify relations in time, we require in addition a standard clock placed at the origin of our Cartesian system of co-ordinates [Pg 16] or frame of reference. If an event takes place anywhere we can assign to it three co-ordinates, , and a time , as soon as we have specified the time of the clock at the origin which is simultaneous with the event. We therefore give an objective significance to the statement of the simultaneity of distant events, while previously we have been concerned only with the simultaneity of two experiences of an individual. The time so specified is at all events independent of the position of the system of co-ordinates in our space of reference, and is therefore an invariant with respect to the transformation (3). It is postulated that the system of equations expressing the laws of pre-relativity physics is co-variant with respect to the transformation (3), as are the relations of Euclidean geometry. The isotropy and homogeneity of space is expressed in this way.[5] We shall now consider some of the more important equations of physics from this point of view. [5]The laws of physics could be expressed, even in case there were a unique direction in space, in such a way as to be co-variant with respect to the transformation (3); but such an expression would in this case be unsuitable. If there were a unique direction in space it would simplify the description of natural phenomena to orient the system of co-ordinates in a definite way in this direction. But if, on the other hand, there is no unique direction in space it is not logical to formulate the laws of nature in such a way as to conceal the equivalence of systems of co-ordinates that are oriented differently. We shall meet with this point of view again in the theories of special and general relativity.