Roberts discovered many properties of geodesic lines and lines of curvature on the ellipsoid, especially in relation to umbilics, and from 1845 published papers in the "Journal de Mathématiques", the "Proceedings of the Royal Irish Academy", "Cambridge and Dublin Mathematical Journal", "Nouvelles Annales de Mathématiques". In 1850 he wrote in the "Journal de Mathématiques" of the lines of curvature and asymptotic lines on a surface, at any point of which the sum of the principal curvatures is zero. The International Exhibition of 1851 at Hyde Park displayed a small model ellipsoid, on which the lines of curvature had been traced according to a method Roberts invented. Roberts published several papers on the properties and functions of the roots of algebraic equations, and on covariants and invariants. From 1868 to 1873 he published work in "Annali di Matematica", including in 1869 and 1871 two papers on Abelian function.