Reading an old book of classical mechanics (Spiegel's theoretical mechanics) I came across a passage that I would like to clarify. In order to find the equations of motion of free falling point particle that regulate motion according to a laboratory on to the earth's surface, the relative acceleration theorem is exploited. In a step (red arrow) apparent gravitational field is introduced to get (6), and then (7) ($\mathbf{r}$ the "small" vector from laboratory origin to the point, the triple vector product has magnitude something like $h\cdot 10^{-9}$). But later (blue arrow) the apparent field is written as $-g\mathbf{k}$ where $\mathbf{k}$ is the laboratory $z$ versor, orthogonal to the ground, toward the sky. Isn't this a bug? The direction of apparent field is not $\mathbf{k}$. Of course the effect is small, but Coriolis force too is small. If this is not a bug, why? If this is a bug, how to fix It?
Edit
The problem is that $\mathbf{g}$ does not represent the real gravitational field but the direction of the plumb line (this is the meaning of the choice in equation (5)). Therefore it has not the direction of the versor $-\mathbf{k}$. The effect is small but the discussion assumes without justifying it that it is negligible compared to that of Coriolis. How can I justify it?
