I consider ordinary three-dimensional space; and use the words point, line, plane, &c., in their ordinary acceptations; only the notion of distance is altered, viz. instead of taking the Absolute to be the circle at infinity, I take it to be a quadric surface: in the analytical developments this is taken to be the imaginary surface x2 + y2 + z2 + w2 = 0, and the formulæ arrived at are those belonging to the so-called Elliptic Space. The object of the Memoir is to set out, in a somewhat more systematic form than has been hitherto done, the general theory; and in particular, to further develop the analytical formulæ in regard to the perpendiculars of two given lines. It is to be remarked that not only all purely descriptive theorems of Euclidian geometry hold good in the new theory; but that this is the case also (only we in nowise attend to them) with theorems relating to parallelism and perpendicularity, in the Euclidian sense of the words. In Euclidian geometry, infinity is a special plane, the plane of the circle at infinity, and we consider (for instance) parallel lines, that is, lines which meet in a point of this plane: in the new theory, infinity is a plane in nowise distinguishable from any other plane, and there is no occasion to consider (although they exist) lines meeting in a point of this plane, that is, parallel lines in the Euclidian sense.