Equation (III-1) applies to the absolute motion of the fluid. It is more practical to deal with the velocity of the fluid relative to a fixed point on the surface of the earth, and to this end the above equation can be transformed by use of Eq. (121), Art. 2.14 to yield an equation involving relative motions only.
As applied to the field of fluid velocity, the transformation equation (121) can be expressed as
where 
is the absolute velocity and
is the relative velocity of that portion of the fluid
which happens to occupy the absolute position 
(or
relative position 
) at the instant t. This equation
is essentially the Eulerian counterpart of Eq. (121) which applies
to a particular mass. The quantities 
and
appearing in Eq. (50) are defined by
The operator 
involves only space derivatives at a given instant
and therefore is independent of any motion of the coordinate system.
Note however that the local time rates of change
and
are
different. The latter is the relative local time rate of change at a
point fixed relative to the earth. The relations (51) may help to shed
further light on the meanings of the absolute and relative
accelerations.
If we consider that the rotation of the earth 
is sensibly
constant then the last term of Eq. (50) vanishes and substitution
of this equation in (III-1) yields the equation of relative motion
on the earth
where
The quantity 
is the earth's gravity wihch was discussed in
article 2.02, and 
and 
are the tide
producing forces per unit mass which were discussed in Articles
2.042 and 2.043.
The term 
, now appearing on the
right side of Eq. (III-3), is the apparent deflecting force associated
with the earth's rotation. This force is directed normal to
the relative velocity vector 
at all times.