The right-hand side is identically equal to the left-hand side on account of the equations of the linear orthogonal transformation, and the right-hand side differs from the left-hand side only in that the are replaced by the . This is expressed by the statement that is an invariant with respect to linear orthogonal transformations. It is evident that in the Euclidean geometry only such, and all such, quantities have an objective significance, independent of the particular choice of [Pg 9] the Cartesian co-ordinates, as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariants, which has to do with the laws that govern the form of invariants, is so important for analytical geometry. As a second example of a geometrical invariant, consider a volume. This is expressed by By means of Jacobi's theorem we may write where the integrand in the last integral is the functional determinant of the with respect to the , and this by (3) is equal to the determinant of the coefficients of substitution, . If we form the determinant of the from equation (4), we obtain, by means of the theorem of multiplication of determinants, If we limit ourselves to those transformations which have the determinant +1,[3] and only these arise from continuous variations of the systems of co-ordinates, then is an invariant. [3]There are thus two kinds of Cartesian systems which are designated as "right-handed" and "left-handed" systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them. [Pg 10] Invariants, however, are not the only forms by means of which we can give expression to the independence of the particular choice of the Cartesian co-ordinates. Vectors and tensors are other forms of expression. Let us express the fact that the point with the current co-ordinates lies upon a straight line. We have Without limiting the generality we can put If we multiply the equations by (compare (3a) and (5)) and sum for all the 's, we get where we have written