The photometric data for 460 classical, fundamental-mode Cepheids in the SMC with log P > 0.4 measured by Udalski et al. have been analyzed for their P-C and P-L relations, and for the variation of amplitude across the instability strip in a similar
way that was done in Papers I and II of this series. The SMC Cepheids are bluer in (B-V) at a given period than for both the Galaxy and the LMC. Their P-C relation in (B-V) is best fit by two lines intersecting at P=10 d. Their break must necessarily exist also in the P-L relations in B and/or V, but remains hidden in the magnitude scatter. An additional pronounced break of the P-L relations in B, V, and I occurs at P=2.5 d. The observed slope of the lines of constant period in the HR diagram agrees with the theoretical expectation from the pulsation equation. The largest amplitude Cepheids for periods less than 13 days occur near the blue edge of the instability strip. The sense is reversed in the period interval from 13 to 20 days, as in the Galaxy and the LMC. The SMC P-L relation is significantly flatter than that for the Galaxy, NGC 3351, 4321, M31, all of which have nearly the same steep slope. The SMC P-L slope is intermediate between that of these steep slope cases and the very shallow slope of Cepheids in the lower metallicity galaxies of NGC 3109 and Sextans A/B, consistent with the premise that the Cepheid P-L relation varies from galaxy-to-galaxy as function of metallicity. Failure to take into account the slope differences in the P-L relation as a function of metallicity using Cepheids as distance indicators results in incorrect Cepheid distances. Part of the 15% difference between our long distance scale - now independently supported by TRGB distances - and that of the HST Key Project short scale is due to the effect of using an inappropriate P-L relation.
