Two systems of eclipse elements are now adopted in the ephemerides and tables; the one, that of F.W. Bessel, is used in the English, American and French ephemerides, the other--P. A. Hansen's--in the German and in the eclipse tables of T. Ritter von Oppolzer. The two have in common certain geometric constructions. The fundamental axis of reference in both systems is the line passing through the centres of the sun and moon; this is the common axis of the shadow cones, which envelop simultaneously the sun and moon as shown in figs. 1, 2, 3. The surface of one of these cones, that of the umbra, is tangent to both bodies externally. This cone comes to a point at a distance from the moon nearly equal to that of the earth. Within it the sun is wholly hidden by the moon. Outside the umbral cone is that of the penumbra, within which the sun is partially hidden by the moon. The geometric condition that the two bodies shall appear in contact, or that the eclipse shall begin or end at a certain moment, is that the surface of one of these cones shall pass through the place of the observer at that moment. Let a plane, which we call the fundamental plane, pass through the centre of the earth perpendicular to the shadow axis. On this plane the centre of the earth is taken as an origin of rectangular co-ordinates. The axis of Z is perpendicular to the plane, and therefore parallel to the shadow axis; that of Y and X lie in the plane. In these fundamental constructions the two methods coincide. They differ in the direction of the axis of Y and X in the fundamental plane. In Bessel's method, which we shall first describe, the intersection of the plane of the earth's equator with the fundamental plane is taken as the axis of X. The axis of Y is perpendicular to it, the positive direction being towards the north. The Besselian elements of an eclipse are then:--x, y, the co-ordinates of the shadow axis on the fundamental plane; d, the declination of that point in which the shadow axis intersects the celestial sphere; [mu], the Greenwich hour angle of this point; l, the radius of the circle, in which the penumbral or outer cone intersects the fundamental plane; and l', the radius of the circle, in which the inner or umbral cone intersects this plane, taken positively when the vertex of the cone does not reach the plane, so that the axis must be produced, and negatively when the vertex is beyond the plane. Hansen's method differs from that of Bessel in that the ecliptic is taken as the fundamental plane instead of the equator. The axis of X on the fundamental plane is parallel to the plane of the ecliptic; that of Y perpendicular to it. The other elements are nearly the same in the two theories. As to their relative advantages, it may be remarked that Hansen's co-ordinates follow most simply from the data of the tables, and are necessarily used in eclipse tables, but that the subsequent computation is simpler by Bessel's method.