This paper reports on extensive spectroscopic observations of the WR-ring nebulae NGC 2359 and RCW 78, respectively, excited by the WN5 stars HD 56925 and WN8 HD 117688. For the first object, abundances were determined for O/H, Ne/H, N/H, and He/H in many different positions, including the ionized gas in the bubble, inside the optical shell structure, and the outermost zones associated with the S 298 H II region. No significant difference in the N/H and O/H abundances was found over the entire nebula. The O/H and N/H abundances expected are close to those for a normal H II region located at similar distance. In the case of He/H, indication is found of local enhancements which sum to the abundance of metal-rich galactic H II regions like M17. RCW 78 appears to show slight overabundances of He/H and N/H in the two observed positions. The ionizing temperature for the WN central star (HD 56925) of NGC 2359 is determined from the observed H II region spectrum giving a value of 50,000 K, appropriate to its spectral type.