Mon.Not.Roy.Astron.Soc. 361 (2005) 460-468 We have used a Fourier Transform Spectrometer on the James Clerk Maxwell Telescope to study the submillimetre continuum emission from dust in three hot molecular cores (HMC). The spectral index beta of the dust emission for these sources has been determined solely within the 30 GHz wide 350 GHz (850 micron) passband to an accuracy comparable to those determined through multi-wavelength observations. We find an average beta = 1.6, in agreement with spectral indices determined from previous submillimetre observations of these sources and with those determined for HMC in general. The largest single source of uncertainty in these results is the variability of the atmosphere at 350 GHz, and with better sky subtraction techniques we show that the dust spectral index can clearly be determined within one passband to high accuracy with a submillimetre FTS. Using an imaging FTS on SCUBA-2, the next generation wide-field submillimetre camera currently under development to replace SCUBA at the JCMT in 2006, we calculate that at 350 GHz it will be possible to determine beta to +/- 0.1 for sources as faint as 400 mJy/beam and to +/- 0.3 for sources as faint as 140 mJy/beam.