Two conics are described having the same focus, and the distance of this focus from the corresponding directrix of each is the same; if the conics touch one another, then twice the sine of half the angle between the transverse axes is equal to the difference of the reciprocals of the eccentricities. If two limaçons are described having the same node and base circles of the same diameter, and if the limaçons touch each other, then twice the sine of half the angle between the axes of the limaçons is equal to the difference of the eccentricities. If a circle of a given radius pass through the focus (S) of a given conic and cut the conic in the points A, B, C, and D; then SA. SB. SC. SD is constant. If a circle of a given radius pass through the node (S) of a given limaçon and cut it in A, B, C, and D; then 1 ——————————————— is constant. (SA. SB. SC. SD) A circle passes through the focus of a conic whose latus rectum is 2l and meets the conic in four points whose distance from the focus are r₁, r₂, r₃, r₄, then 1 1 1 1 2 ——— + ——— + ——— + ——— = ——— . r₁ r₂ r₃ r₄ l A circle passes through the node of a limaçon whose latus rectum is 2l, meeting the curve in four points whose distances from the node are r₁, r₂, r₃, r₄, then r₁ + r₂ + r₃ + r₄ = 2l. Two points P and Q are taken, one on each of two conics which have a common focus and their axes in the same direction, such that PS and QS are at right angles, S being the common focus. Then the tangents at P and Q meet on a conic the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original conics. Two points P and Q are taken one on each of two limaçons which have a common node and their axes in the same direction, such that PS and QS are at right angles, S being the common node. Then the nodal tangent circles at P and Q intersect on a limaçon the square of whose eccentricity is equal to the sum of the squares of the eccentricities of the original limaçons. A series of conics are described with a common latus rectum; the locus of points upon them at which the perpendicular from the focus on the tangent is equal to the semi-latus rectum is given by the equation