Hamilton, in his Elements of Quaternions (Book I. Chapter II.), developed this system of coordinates, by means of his vectorial algebra, in connection with his theory of geometrical nets. In the following paper I have re-proved the fundamental theorems, by other methods, and have sketched out the natural lines of development of the system. Though few, if any, of the results arrived at are new, they are at any rate newly stated; and I believe that the elegance of method will command interest.