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In this paper we confirm the existence of period-luminosity (PL) andperiod-luminosity-colour (PLC) relations at maximum light for O and C Miravariables in the LMC. We demonstrate that in the J and H bands the maximumlight PL relations have a significantly smaller dispersion than theircounterparts at mean light, while the K band and bolometric PL relations have adispersion comparable to that at mean light. In the J, H and K bands the fittedPL relations for the O Miras are found to have smaller dispersion than thosefor the C Miras, at both mean and maximum light, while the converse is true forthe relations based on bolometric magnitudes. The inclusion of a non-zero logperiod term is found to be highly significant in all cases except that of the CMiras in the J band, for which the data are found to be consistent with havingconstant absolute magnitude. This suggests the possibility of employing C Mirasas standard candles.  We suggest both a theoretical justification for the existence of Mira PLrelations at maximum light and a possible explanation of why these relationsshould have a smaller dispersion than at mean light. The existence of suchmaximum light relations offers the possibility of extending the range andimproving the accuracy of the Mira distance scale to Galactic globular clustersand to other galaxies.Comment: 17 pages, Latex. Accepted for publication in MNRA

Period-luminosity and period-luminosity-colour relations for Mira variables at maximum light