  Over 200 years ago, Gauss discovered a composition law on the $SL_2({\mathbb
Z})$-equivalence classes of primitive binary quadratic forms. Since then,
bijections of classes of binary forms have been found with ideal class groups
of quadratic rings. This paper uses one such bijection given by Bhargava,
relating classes of projective binary cubic forms to the $3$-torsion of an
ideal class group, to find an explicit form for a cubic analogue of Gaussian
composition.
