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The photometric data for 460 classical, fundamental-mode Cepheids in the SMCwith log P > 0.4 measured by Udalski et al. have been analyzed for their P-Cand P-L relations, and for the variation of amplitude across the instabilitystrip in a similar way that was done in Papers I and II of this series. The SMCCepheids are bluer in (B-V) at a given period than for both the Galaxy and theLMC. Their P-C relation in (B-V) is best fit by two lines intersecting at P=10d. 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 ofthe P-L relations in B, V, and I occurs at P=2.5 d. The observed slope of thelines of constant period in the HR diagram agrees with the theoreticalexpectation from the pulsation equation. The largest amplitude Cepheids forperiods 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 theGalaxy and the LMC. The SMC P-L relation is significantly flatter than that forthe 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 andthe very shallow slope of Cepheids in the lower metallicity galaxies of NGC3109 and Sextans A/B, consistent with the premise that the Cepheid P-L relationvaries from galaxy-to-galaxy as function of metallicity. Failure to take intoaccount the slope differences in the P-L relation as a function of metallicityusing Cepheids as distance indicators results in incorrect Cepheid distances.Part of the 15% difference between our long distance scale - now independentlysupported by TRGB distances - and that of the HST Key Project short scale isdue to the effect of using an inappropriate P-L relation.

New period-luminosity and period-color relations of classical Cepheids: III. Cepheids in SMC