Substitutional equations of the type considered by the late Alfred Young are shown to be intimately related with the theory of idempotents. Any equation LX = o possessing solutions other than X = o is shown to have the same solutions as another equation MX = o, where M is obtained from L by premultiplying the latter by a suitably chosen expression A and where the minimum equation of M is xψ(x) = o, ψ(x) being prime to x. The expression ψ(M) is then idempotent, and it is shown that the most general solution of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is X = ψ(M)Y, where Y is an arbitrary expression. The number of linearly independent solutions of LX = o is kn!, where k is the coefficient of the unit permutation in ψ(M) when that expression is expressed in terms of the permutations of the symmetric group Sn.

Corresponding results are obtained for the equation LX = R, and methods are given for solving sets of simultaneous equations of both types.