The following pages contain, in a comparatively compact form, part of the substance of a voluminous paper read to the Society six years ago. Of that paper, which employed ordinary analysis alone, only a few pages had been put in type when I succeeded in overcoming a formidable difficulty which had presented itself in my quaternion treatment of the subject. I therefore withdrew the paper in order that it might have the benefit of the simplification which quaternions always give; but it is only of late that I have found time to complete part of the translation into the new language. From the circumstances under which the paper has thus been produced, i, j , k come forward with undue prominence, a thing to be regarded (in Hamilton's words) “ as an inelegance and imperfection in this calculus, or rather in the state to which it has hitherto been unfolded.” Immense as is the simplification already attained, it is clear that in many places still more is attainable. But I have not postponed my paper till it should receive this final polish, partly because the time I can devote to such inquiries is extremely limited, and partly because I think that several of the results obtained, and of the modes of obtaining them, are new and remarkable. Besides, a question of this order of difficulty is admirably adapted to show in what respects quaternion methods require improvement. There must be some simple mode of deducing (13) and (21) below from (7) without the explicit use of i, j , k, but I have not yet been fortunate enough to discover it.