Integrating the radial part of the Fourier transform of the product of N hydrogenic orbitals results in an associated Legendre function that can be reduced to a finite series of elementary functions. This transform is found to depend on a polynomial in the wave vector k divided by a binomial in k2 raised to a power that is the sum of principle quantum numbers. This form facilitates the analytical reduction of integrals arising from orthogonalization corrections in atomic processes. Transforms for the product of orbital pairs (1s,1s) through (1s,3d) are given explicitly.