be the contra-variant energy-tensor of the liquid. To it also belongs
the covariant tensor
(58a) T_{μν} = -_g__{μν} _p_ + _g__{μα} _dx__{α}/_ds_ _g__{μβ}
_dx__{β}/_ds_ ρ
as well as the mixed tensor
(58b) T^α_{σ} = -δ^α_{σ} _p_ + _g__{σβ} _dx__{β}/_ds_ _dx__{α}/_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
_g__{αβ} _dx__α/_ds_ _dx__{β}/_ds_ = 1,
are sufficient, with the given values of _g__{αβ}, for finding out the
six unknowns
_p_, ρ, _dx₁_/_ds_, _dx₂_/_ds_, _dx₃_/_ds_ _dx₄_/_ds_.
If _g__{μν}’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 φ_{ν} be the components of a covariant four-vector, the
electro-magnetic potential; from it let us form according to (36) the
components F_{ρσ} of the covariant six-vector of the electro-magnetic
field according to the system of equations
(59) F_{ρσ} = ∂φ_{ρ}/∂_x__{σ} - ∂φ_{σ}/∂_x__{ρ}.