p = -r cos 2_x_ If a series of limaçons are described with the same latus rectum, the locus of points upon them at which the diameter of the nodal tangent circle is equal to the semi-latus rectum, is given by the equation pr = -cos 2_x_ If POP₁ be a chord of a conic through a fixed point O, then will tan ½P₁SO tan ½PSO be a constant, S being the focus of the conic. If POP₁ be a nodal circle of a limaçon passing through a fixed point O, then will tan ½ P₁SO tan ½ PSO be a constant, S being the node. Conics are described with equal latera recta and a common focus. Also the corresponding directrices envelop a fixed confocal conic. Then these conics all touch two fixed conics, the reciprocals of whose latera recta are the sum and difference respectively of those of the variable conic and their fixed confocal, and which have the same directrix as the fixed confocal. Limaçons are described with equal latera recta and a common node. Also the director circles envelop a fixed limaçon having a common node. Then these limaçons all touch two fixed limaçons whose latera recta are the sum and difference respectively of the reciprocals of the variable limaçon and of the fixed limaçon, and which have the same base circle as the fixed limaçon. Every focal chord of a conic is cut harmonically by the curve, the focus, and the directrix. Every nodal chord of a limaçon is bisected by the base circle. The envelope of circles on the focal radii of a conic as diameters is the auxiliary circle. The envelope of the perpendiculars at the extremities of the nodal radii of a limaçon is a circle having for the diameter the axis of the limaçon. Below we give a number of theorems respecting the cardioid obtained by inverting the corresponding theorems concerning the parabola. The straight line which bisects the angle contained by two lines drawn from the same point in a parabola, the one to the focus, the other perpendicular to the directrix, is a tangent to the parabola at that point.