Throughout his career mathematics remained a major interest for Harley. He spent time on the theory of equations, especially the theory of the general equation of the fifth degree. His conclusions, which were published in Memoirs of the Manchester Lit. and Phil. Soc 1860, xv. 172–219, were independently reached at the same time by Sir James Cockle. Harley's two further papers on the Theory of QuinticsQuarterly Journal of Mathematics 1860–2, iii. 343–59; v. 248–60 and an exposition of Cockle's method of symmetric products in Phil. Trans. (1860) attracted the attention of Arthur Cayley, who carried the research further.
