I was originally attracted to the study of Quaternions by Sir W. R Hamilton's ingeniously devised and most valuable operator

to which he called special attention (Lectures on Quaternions, § 620) on account of its promise of usefulness in physical applications. But I soon found that in order that its full power may be applied, in general investigations, it is necessary that we should have processes of definite integration, of the kinds required in physics, applicable to quaternion symbols and not merely to scalar variables. I often consulted Hamilton about this want, and he promised to endeavour to supply it at some future time. I fancy that shortly before his death he must have in some way supplied it, though he certainly did not print, nor does he appear even to have written, anything on the subject. In one of the last letters I received from him, he said that he intended to conclude the final chapter of his Elements, which is devoted to physical applications, by some sections on the use of the operator mentioned above. That chapter remains unfinished, and as Hamilton rarely wrote down the steps of even a complex train of mathematical reasoning until he had mentally completed it, it is to be feared that this portion of his investigations is entirely lost. So far as the analytical aspect of Quaternions is concerned, this loss is very serious indeed, for there can be little doubt that Hamilton's solution would have been of immense value from the purely mathematical point of view.