**Mathematical and Scientific Investigations**

Cockle is also remembered for his mathematical and scientific investigations. For instance he invented the number systems of tessarines and coquaternions, and worked with Arthur Cayley (1821–1895) on the theory of linear algebra. Like many young mathematicians he attacked the problem of solving the quintic equation, notwithstanding Abel-Ruffini theorem that a solution by radicals was impossible. In this field Cockle achieved some notable results, amongst which is his reproduction of Sir William R. Hamilton's modification of Abel's theorem. Algebraic forms were a favourite object of his studies. He also made contributions to the theory of differential equations, in particular the development of the theory of differential invariants or criticoids.

He displayed a keen interest in scientific societies. From 1863 to 1879 he was president of the Queensland Philosophical Society (now incorporated in the Royal Society of Queensland); on his return to England he became associated with the London Mathematical Society, of which he was president from 1886 to 1888, and the Royal Astronomical Society, serving as a member of the council from 1888 to 1892. He died in London on the 27th of January 1895.

An obituary notice by the Revd. Robert Harley was published in 1895 in *Proc. Roy. Soc.* vol. 59. A volume containing his scientific and mathematical researches made during the years 1864–1877 was presented to the British Museum in 1897 by his widow. Like his father, Sir James became extremely wealthy during his lifetime, leaving an estate of £32,169, which is approximately £2.7 million inflation adjusted as of 2008.

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