Summary: We establish the equations of motion for a fluid spherical shell which constitutes the upper-most layer of a rotating body and which is heated from below because of radiogenic decay occurring in the lower layers. We solve these equations in terms of series of products of Bessel functions and Gegenbauer polynomials. By the stepwise solution of an infinite-order determinant equation we determine those values for the Taylor and Rayleigh numbers of the flow, up to the \(10^{16}\) order, that will give rise to convective cells in the liquid layer. Using these eigenvalues, we solve linear systems of equations to determine the coefficients of the series solutions.