Page:The Meaning of Relativity - Albert Einstein (1922).djvu/23

Rh where the integrand in the last integral is the functional determinant of the $$x_\nu'$$ with respect to the $$x_\nu$$, and this by (3) is equal to the determinant $$|b_{\mu \nu}|$$ of the coefficients of substitution, $$b_{\nu\alpha}$$. If we form the determinant of the $$\delta_{\mu\alpha}$$ 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, and only these arise from continuous variations of the systems of co-ordinates, then $$V$$ is an invariant.

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 $$x_\nu$$ lies upon a straight line. We have

Without limiting the generality we can put

If we multiply the equations by $$b_{\beta \nu}$$ (compare (3a) and (5)) and sum for all the $$\nu$$'s, we get