averaged over $8$ independent realizations of the random disorder, vs disorder strength $p$, for the system of size $20\times20\times 6$. As $p$ increases we see that $[\Delta E]$ decreases and appears to vanish. A vanishing $[\Delta E]$ implies a vanishing entropy jump and hence a vanishing of the delta-function specific heat singularity usually associated with the first order melting transition. Such a vanishing of the specific heat singularity, observed experimentally on increasing applied magnetic field at fixed disorder strength, has been used as evidence for a weakening of the first order melting transition and its termination at a second order upper critical point.[@Bouquet] However, if the first order phase transition is indeed to vanish at a second order critical point, it is necessary that the discontinuities in [*all*]{} thermodynamic first derivatives of the free energy vanish as the critical point is approached. In Fig. \[f5\]a, however, we see that the discontinuity $[\Delta Q]$ [*increases*]{} as $p$ increases, and does not vanish as it must if the first order line is to end in a critical point. We thus find that our melting transition remains strongly first order for all disorder strengths $p$. The vanishing of $[\Delta E]$ upon increasing $p$ 