While engaged in some quaternion researches with reference to Spherical Harmonics, which I hope soon to lay before the Society, I was led to imagine that some of my results might produce a simplification of the ordinary modes of treating the subject. The following is the result of the attempt. It seems to make the calculus somewhat more intelligible to the beginner than the methods employed by O'Brien and Murphy, whose works on the subject are usually read in this country. As I am not writing a treatise, but merely sketching a method, I shall run over the principal elementary propositions only.