These are the equations of straight lines with respect to a second Cartesian system of co-ordinates '. They have the same form as the equations with respect to the original system of co-ordinates. It is therefore evident that straight lines have a significance which is independent of the system of co-ordinates. Formally, this depends upon the fact that the quantities () - are transformed as the components of an interval, . The ensemble of three quantities, defined for every system of Cartesian co-ordinates, and which transform as the components of an interval, is called a vector. If the three [Pg 11] components of a vector vanish for one system of Cartesian co-ordinates, they vanish for all systems, because the equations of transformation are homogeneous. We can thus get the meaning of the concept of a vector without referring to a geometrical representation. This behaviour of the equations of a straight line can be expressed by saying that the equation of a straight line is co-variant with respect to linear orthogonal transformations. We shall now show briefly that there are geometrical entities which lead to the concept of tensors. Let be the centre of a surface of the second degree, any point on the surface, and the projections of the interval upon the co-ordinate axes. Then the equation of the surface is In this, and in analogous cases, we shall omit the sign of summation, and understand that the summation is to be carried out for those indices that appear twice. We thus write the equation of the surface The quantities determine the surface completely, for a given position of the centre, with respect to the chosen system of Cartesian co-ordinates. From the known law of transformation for the (3a) for linear orthogonal transformations, we easily find the law of transformation for the :[4] [4]The equation may, by (5), be replaced by = 1, from which the result stated immediately follows. [Pg 12] This transformation is homogeneous and of the first degree in the . On account of this transformation, the , are called components of a tensor of the second rank (the latter on account of the double index). If all the components, , of a tensor with respect to any system of Cartesian co-ordinates vanish, they vanish with respect to every other Cartesian system. The form and the position of the surface of the second degree is described by this tensor ().