Observations made at the quiet Sun-centre with the Coronal Diagnostic Spectrometer (CDS) and Solar Ultraviolet Measurements of Emitted Radiation (SUMER) instruments on the Solar and Heliospheric Observatory (SOHO) have shown that the intensities of the resonance lines of HeI and HeII are significantly larger than predicted by emission measure distributions found from other transition region lines. The intensities of the helium lines are observed to be lower in coronal holes than in the quiet Sun. Any theory proposed to account for the behaviour of the helium lines must explain the observations of both the quiet Sun and coronal holes. We use observations made with SOHO to find the physical conditions in a polar coronal hole. The electron pressure is found using the CIII 1175-Å and NIII 991.5-Å lines, as the CIII line at 977.0Å becomes optically thick in some regions at high latitudes. The mean electron pressure is a factor of ~=2 lower than that at the quiet Sun-centre. The mean coronal electron temperature is <= 9.4×105K. The helium lines are enhanced with respect to other transition region lines but by factors which are ~=30 per cent smaller than at the quiet Sun-centre. The mean ratios of the intensities of the HeI 537.0- and 584.3-Å lines and of the HeI and HeII 303.8-Å lines vary little with the type of region studied. These ratios are compared with those predicted by models of the transition region, taking into account the radiative transfer in the helium lines. No significant variation is found in the relative abundances of carbon and silicon.