We present new spectroscopic observations of 13 H II regions in the Local Group spiral galaxy M33. The regions observed range from 1 to 7 kpc in distance from the nucleus. Of the 13 H II regions observed, the [O III] 4363 Angstrom line was detected in six regions. Electron temperatures were thus able to be determined directly from the spectra using the [O III] 4959,5007 A/4363 A line ratio. Based on these temperature measurements, oxygen and neon abundances and their radial gradients were calculated. For neon, a gradient of -0.016 +/- 0.017 dex/kpc was computed, which agrees with the Ne/H gradient derived previously from ISO spectra. A gradient of -0.012 +/- 0.011 dex/kpc was computed for O/H, much shallower than was derived in previous studies. The newly calculated O/H and Ne/H gradients are in much better agreement with each other, as expected from predictions of stellar nucleosynthesis. We examine the correlation between the WC/WN ratio and metallicity, and find that the new M33 abundances do not impact the observed correlation significantly. We also identify two new He II-emitting H II regions in M33, the first to be discovered in a spiral galaxy other than the Milky Way. In both cases the nebular He II emission is not associated with Wolf-Rayet stars. Therefore, caution is warranted in interpreting the relationship between nebular He II emission and Wolf-Rayet stars when both are observed in the integrated spectrum of an H II region.
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