Page:The principle of relativity (1920).djvu/191

 Similarly

B^{μν} = g^{μν}g_{αβ}A^{αβ}.

It is to be remarked that g^{μν} is no other than the "complement" of g_{μν} for we have,—

g^{μα}g^{νβ}g_{αβ} = g_{μα}δ^ν_α = g^{μν}.

§ 9. Equation of the geodetic line (or of point-motion).

As the "line element" ds is a definite magnitude independent of the co-ordinate system, we have also between two points P_{1} and P_{2} of a four dimensional continuum a line for which [integral]ds is an extremum (geodetic line), i.e., one which has got a significance independent of the choice of co-ordinates.

Its equation is

(20) δ{ [integral]^{P_{2}}_{P_{1}} ds } = 0

From this equation, we can in a wellknown way deduce 4 total differential equations which define the geodetic line; this deduction is given here for the sake of completeness.

Let λ, be a function of the co-ordinates x_{ν}; This defines a series of surfaces which cut the geodetic line sought-for as well as all neighbouring lines from P_{1} to P_{2}. We can suppose that all such curves are given when the value of its co-ordinates x_{ν} are given in terms of λ. The