The following quaternion notes are interesting as the last piece of mathematical work written down by Professor Tait. During his last severe illness he referred several times to the importance of working out still further the theory of the linear vector function; but he had not been able to put pen to paper for months previous to July 2, 1901, just two days before his death. On that date he wrote down the notes now printed, probably at intervals throughout the day. They would serve, in all likelihood, chiefly as nuclei round which his thoughts would circle; and they will now serve to indicate in some measure the directions in which his thoughts were moving. The foolscap sheet on which the notes were written was handed to his eldest son, with the request that it should be preserved, as he believed it contained the germ of an important advance. A facsimile of this sheet, on a reduced scale, is reproduced in the accompanying Plate.