Oceanic transform faults are one of the main types of plate boundary, but the manner in which they slip remains poorly understood. Early studies suggested that relatively slow earthquake rupture might be common; moreover, it has been reported that very slow slip precedes some oceanic transform earthquakes, including the 1994 Romanche earthquake. The presence of such detectable precursors would have obvious implications for earthquake prediction. Here we model broadband seismograms of body waves to obtain well-resolved depths and rupture mechanisms for 14 earthquakes on the Romanche and Chain transform faults in the equatorial Atlantic Ocean. We found that earthquakes on the longer Romanche transform are systematically deeper than those on the neighbouring Chain transform. These depths indicate that the maximum depth of brittle failure is at a temperature of approximately 600 degrees C in oceanic lithosphere. We find that the body waves from the Romanche 1994 earthquake can be well modelled with relatively deep slip on a single fault, and we use the mechanism and depth of this earthquake to recalculate its source spectrum. The previously reported slow precursor can be explained as an artefact of uncertainties in the assumed model parameters.