The Laplace relation and the masses of Uranus' satellites

The theory of the effect of Ariel and Umbriel on Miranda's orbit is completed with a generalization of Souillart's theory of the Laplace relation. Comparison of observations of Miranda's motion with the theory yields an upper limit for the product of Ariel and Umbriel's masses of about 10^?9, where Uranus' mass is unity. Therefore the albedos of Ariel and Umbriel cannot both be as low as the albedos of the darkest asteroids.