A homogeneous ternary cubic equation can in general be taken by a real transformation into the canonical form \[ ax^3+ by^3 + cz^3 = dxyz.\tag{1} \] The author considers the possibility of doing so by a rational transformation; and for certain classes of equations, involving two or more parameters, he gives simple necessary and sufficient conditions. He goes on to the problem of deciding whether or not (1) is soluble (with \(xyz\neq 0)\). General methods for doing so are explained, and the results are given in a large number of special cases (in tabular form). In particular, the case \(a=b=c\) of (1) is disposed of for \(-81 \leq d\leq 80\).