Hamilton, Lectures on Quaternions (Dublin, 1853), Elements of Quaternions (ibid., 1866); H.
His first great work, Lectures on Quaternions (Dublin, 1852), is almost painful to read in consequence of the frequent use of italics and capitals.
The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative.
For (3) (a) we are constrained to refer the reader to Joly's own Manual of Quaternions (1905).
Here the symmetry points at once to the selection of the three principal axes as the directions for i, j, k; and it would appear at first sight as if quaternions could not simplify, though they might improve in elegance, the solution of questions of this kind.
Had quaternions effected nothing more than this, they would still have inaugurated one of the most necessary, and apparently impracticable, of reforms.
He was the author of two text-books on them - one an Elementary Treatise on Quaternions (1867), written with the advice of Hamilton, though not published till after his death, and the other an Introduction to Quaternions (1873), in which he was aided by Professor Philip Kelland (1808-1879), who had been one of his teachers at Edinburgh.
His earliest work dealt mainly with mathematical subjects, and especially with quaternions (q.v.), of which he may be regarded as the leading exponent after their originator, Hamilton.