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tokens: (King 2008 , 2009 ). Moreover, we assumed L L X for the sake of simplicity, but in fact the bolometric correction of the inferred X-ray luminosity might vary from system to system. This might depend e.g. on the line of sight through the outflow, as hinted by the change in spectral hardness among the ULXs considered in Table 1 , as well as by the conjectured connection with ultraluminous supersoft sources (Middleton etal. 2011 , 2015 ; Pinto etal. 2017 ). According to this scenario, NGC 1313 X-1 is likely to be seen more face-on because of the harder spectrum, and therefore L L X is a reasonable assumption (but it is more likely to suffer from beaming though), while NGC 55 ULX and NGC 5408 X-1 have softer spectra that may come from the reprocessing of the harder radiation from the inner edge of the disc (and perhaps by a close hot corona) by the optically thick wind, for which L X L . This latter correction might partially compensate the effect of beaming, but note that L X as well is assumed isotropic and neglects beaming effects. None the less, even when reduced by a factor 23, our inferred masses are consistent with previous estimates of similar systems as well as with the aforementioned scenario of ULX formation. Our calculations also neglect general relativistic effects, such as spinning BHs, and magnetic fields. We explored the impact of a spinning BH by changing the inner boundary condition and the normalizing radiative efficiency . We find values of M up to 20percent lower than in the no spin cases, owing to larger values of and at similar accretion rates. Magnetic fields might also contribute to accelerate the outflow (Blandford Payne 1982 ), effectively lowering w and increasing L for the same accretion rate. This might also lower the value of the inferred central mass. Regardless of their masses, BHs are not the only possible accretors powering ULXs. Indeed, the light curves of three ULXs show sinusoidal pulses with a period 1s that are a distinctive signature of neutron stars (Bachetti etal. 2014 ; Frst etal. 2016 ; Israel etal. 2017 ). Recently, King Lasota ( 2016 ) and King, Lasota Kluzniak ( 2017 ) argued that neutron stars may actually power a fraction of ULXs larger than previously expected. According to their analysis, pulsations may only be observed during a rather short phase of the ULX evolution, preventing an easy detection. However, the variability of the optical spectrum has revealed the existence of a massive stellar BH, likely 2030M , in M101 ULX-1 (Liu etal. 2013 ) and perhaps also in X-ray binaries associated with WolfRayet stars (Prestwich etal. 2007 ; Crowther etal. 2010 ; but see also Laycock etal. 2015 ). Therefore, it appears evident that ULXs comprise a diverse variety of accreting objects, and whether the majority is represented by BHs or neutron stars must be scrutinized further. Clearly, our analysis can be applied only to the subset of ULXs with outflows that probably host an accreting BH; none the less,