In this paper, we derive a physical argument for the existence of Period-luminosity and period-luminosity-colour relations at maximum light. We examine in detail a sample of Cepheids in the Large Magellanic Cloud, and compare the variance of some PL and PLC type distance indicators based on mean and maximum light. We show that a PLC relation based on maximum light leads to a distance estimator with a dispersion about $10 %$ smaller than its counterpart using mean light. We also show that a PLC type relation constructed using observations at both maximum and mean light has a significantly $( > 50 %)$ smaller dispersion than a PLC relation using either maximum or mean light alone. A comparable $( > 30 %)$ reduction in the dispersion of the corresponding distance estimator, however, in this case requires the relation be applied to a large $( n > 30)$ group of equidistant Cepheids in, e.g., a distant galaxy. Recent HST observations of IC4182, M81 and M100 already provide suitable candidate data sets for this relation.
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