  A graphical and an algebraic demonstration is made to show why the slope and
zero point of the Cepheid period-luminosity (P-L) relation is rigidly coupled
with the slope and zero point of the Cepheid instability strip in the HR
diagram. The graphical demonstration uses an arbitrary (toy) ridge line in the
instability strip, while the algebraic demonstration uses the pulsation
equation into which the observed P-L relations for the Galaxy and the LMC are
put to predict the temperature zero points and slopes of the instability
strips. Agreement between the predicted and measured instability strip slopes
argue that the observed P-L differences between the Galaxy and LMC are real. In
another proof, the direct evidence for different P-L slopes in different
galaxies is shown by comparing the Cepheid data in the Galaxy, the combined
data in NGC 3351 and NGC 4321, in M31, LMC, SMC, IC 1613, NGC 3109, and in
Sextans A+B. The P-L slopes for the Galaxy, NGC 3351, NGC 4321, and M31 are
nearly identical and are the steepest in the sample. The P-L slopes decrease
monotonically with metallicity in the order listed, showing that the P-L
relation is not the same in different galaxies, complicating their use in
calibrating the extragalactic distance scale.
