It has long been recognized that the finite speed of light can affect the observed time of an event. For example, as a source moves radially toward or away from an observer, the path length and therefore the light travel time to the observer decreases or increases, causing the event to appear earlier or later than otherwise expected, respectively. This light travel time effect (LTTE) has been applied to transits and eclipses for a variety of purposes, including studies of eclipse timing

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... ns (ETVs) and transit timing variations (TTVs) that reveal the presence of additional bodies in the system. Here we highlight another non-relativistic effect on eclipse or transit times arising from the finite speed of light---caused by an asymmetry in the transverse velocity of the two eclipsing objects, relative to the observer. This asymmetry can be due to a non-unity mass ratio or to the presence of external barycentric motion. Although usually constant, this barycentric and asymmetric transverse velocities (BATV) effect can vary between sequential eclipses if either the path length between the two objects or the barycentric transverse velocity varies in time. We discuss this BATV effect and estimate its magnitude for both time-dependent and time-independent cases. For the time-dependent cases, we consider binaries that experience a change in orbital inclination, eccentric systems with and without apsidal motion, and hierarchical triple systems. We also consider the time-independent case which, by affecting the primary and secondary eclipses differently, can influence the inferred system parameters, such as the orbital eccentricity.