The following paper contains a short digest of investigations communicated to the Society on several occasions during the past, and the present, session. The work had been for some months laid aside, but my attention was recalled to it by Professor Chrystal's valuable paper, in which he treats Minding's Theorem as an example of Plücker's methods, and also by the help of Rodrigues' co-ordinates. I am induced to publish a few of my results in full, as I think that a comparison of the analysis employed by Chrystal, with the very different analysis employed by myself, may be useful as well as interesting, especially from the point of view of the simplicity of the quaternion method. Even when the quaternion processes are written out at full length, they are in general shorter than the most condensed forms of ordinary analysis; and there can be no doubt that they are much more easily interpretable into the corresponding geometrical ideas.