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
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.