A simple analysis is presented for the finite-amplitude, steady motions in a rotating layer of fluid which is heated uniformly from below and cooled from above. The boundaries are considered to be ‘free’ and a solution is obtained for the two-dimensional problem using the eigenfunctions of the stability problem plus the smallest number of higher modes required to represent non-linear interactions. In his analysis of the stability problem Chandrasekhar (1953) concluded that in mercury overstable motions can occur for a value of the Rayleigh number which is as little as 1/67 of the value required for instability to steady motions. In the present paper it is shown that, for a restricted range of Taylor number, steady finite-amplitude motions can exist for values of the Rayleigh number smaller than the critical value required for overstability. The horizontal scale of these finite-amplitude steady motions is larger than that of the overstable motions. A more exact solution to the finite-amplitude problem confirms the above results. The latter solution together with additional physical results will be presented in a later paper.