Lame's equations ´ The directions in which X-rays are strongly scattered by a crystal are determined from the solutions x of Lame's equations, namely
x·a = L, x·b = M, x·c = N,
where {a, b, c} are the basis vectors of the crystal lattice, and L, M, N are any integers. Show that the solutions of Lame's equations are
x = La* + Mb* + Nc*,
where {a∗, b∗, c∗} is the reciprocal basis to {a, b, c}.