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A graphical and an algebraic demonstration is made to show why the slope andzero point of the Cepheid period-luminosity (P-L) relation is rigidly coupledwith the slope and zero point of the Cepheid instability strip in the HRdiagram. The graphical demonstration uses an arbitrary (toy) ridge line in theinstability strip, while the algebraic demonstration uses the pulsationequation into which the observed P-L relations for the Galaxy and the LMC areput to predict the temperature zero points and slopes of the instabilitystrips. Agreement between the predicted and measured instability strip slopesargue that the observed P-L differences between the Galaxy and LMC are real. Inanother proof, the direct evidence for different P-L slopes in differentgalaxies is shown by comparing the Cepheid data in the Galaxy, the combineddata in NGC 3351 and NGC 4321, in M31, LMC, SMC, IC 1613, NGC 3109, and inSextans A+B. The P-L slopes for the Galaxy, NGC 3351, NGC 4321, and M31 arenearly identical and are the steepest in the sample. The P-L slopes decreasemonotonically with metallicity in the order listed, showing that the P-Lrelation is not the same in different galaxies, complicating their use incalibrating the extragalactic distance scale.

Temperature Differences in the Cepheid Instability Strip Require Differences in the Period-Luminosity Relation in Slope and Zero Point