In this paper an attempt is made to supply, at least in part, what the author has long felt as a want in the beautiful system of quaternions, so far as it has yet been developed. To apply it to general inquiries connected with electricity, fluid motion, &c, we require to have means of comparing quaternion-integrals taken over a closed surface with others extended through the enclosed space—and of comparing integrals taken over a non-closed surface with others extended round its boundary.