II. On the equilibrium of rotating liquid cylinders

1. As a preliminary to attacking the problem oi determining the equilibrium configurations of a rotating mass of liquid, I was led to consider whether some method could not be devised for calculating the potential of a homogeneous mass in a manner more simple than that usually adopted. What was obviously required was a calculus enabling us to write down the potential of such a mass by an algebraical transformation of the equation of its boundary, instead of by an integration extending throughout its volume. There was found to be no difficulty in reducing the calculation to a problem of algebraical transformation, but in three-dimensional problems the transformations required were, in general, as impracticable as the integrations which they were intended to replace. This was because the transformations depended upon a continued application of the formula which expresses the products or powers of spherical harmonics as the sum of a series of harmonics.