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 period-color ( P-C ) and period-luminosity ( 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 ) ^ { 0 } at a given period than for both the Galaxy and the LMC .
Their P-C relation in ( B - V ) ^ { 0 } is best fit by two lines intersecting at P = 10 days .
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 days .
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 , NGC 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 tip of the red-giant branch ( TRGB ) distances – and that of the HST Key Project short scale is due to the effect of using an inappropriate P-L relation .