This paper is intended as a necessary supplement to the paper bearing the same title already printed in the Society's Transactions, though but an abridgment of a larger one which the author had prepared on the subject. Particular circumstances induced him to alter the plan he had originally contemplated, and instead of a complete development in detail of his researches and his views, he has only on the present occasion given so much of his results as were necessary to bring the system of polar spherical co-ordinates to a state analogous to that in which plane polar curves has long been stationary, one point of the analogy excepted, viz. where the author has extended the method of treating tangents and normals, and the consequent investigations dependent on these, by giving the polar equations of those lines, instead of merely examining the relation between the radius-vector, and perpendicular upon the tangent. In a note the equations of the tangent and normal, to plain curves, is given from first principles; and the analogy between plane and spherical curves is shewn to be remarkably close.