The method of invariants is an approach to the problem of reconstructing the phylogenetic tree of a collection of $m$ taxa using nucleotide sequence data. Models for the collection of probabilities of the $4^m$ possible vectors of bases at a given site will have unknown parameters that describe the random mechanism by which substitution qoccurs along the branches of a possible phylogenetic tree. An invariant is a polynomial in these probabilities that, for a given phylogeny, is zero for all choices of the substitution mechanism parameters. We show for a widely used, general class of substitution mechanisms that given two different trees there is always a polynomial that is an invariant for one tree but not an invariant for the other. Thus estimates of invariants can always be used to discriminate between competing phylogenies.