(58b) Tασ = -δασ p + gσβ dxβ/ds dxα/ds ρ.
If we put the right-hand side of (58b) in (57a) we get the general hydrodynamical equations of Euler according to the generalised relativity theory. This in principle completely solves the problem of motion; for the four equations (57a) together with the given equation between p and ρ, and the equation
gαβ dx_α/ds dxβ/ds = 1,
are sufficient, with the given values of gαβ, for finding out the six unknowns
p, ρ, dx₁/ds, dx₂/ds, dx₃/ds dx₄/ds.
If gμν’s are unknown we have also to take the equations (53). There are now 11 equations for finding out 10 functions g, so that the number is more than sufficient. Now it is be noticed that the equation (57a) is already contained in (53), so that the latter only represents (7) independent equations. This indefiniteness is due to the wide freedom in the choice of co-ordinates, so that mathematically the problem is indefinite in the sense that three of the space-functions can be arbitrarily chosen.
§20. Maxwell’s Electro-Magnetic field-equations.
Let φν be the components of a covariant four-vector, the electro-magnetic potential; from it let us form according to (36) the components Fρσ of the covariant six-vector of the electro-magnetic field according to the system of equations
(59) Fρσ = ∂φρ/∂xσ - ∂φσ/∂xρ.
From (59), it follows that the system of equations