A series of conics through four fixed points is cut by any transversal in a range of points in involution. Inverting and projecting:—a series of connodal trinodal quartics can be passed through four other fixed points; any conic through the three nodes cuts the series of quartics in pairs of points which determine at a node a pencil in involution. The conic touches two of the quartics and the lines to the points of contact are the foci of the pencil. If the sides of two triangles touch a given conic, their six angular points will lie on another conic. Inverting and projecting:—if two groups of three conics each be passed through three nodes and tangent to the quartic, their six points of intersection (three of each group) lie on another connodal trinodal quartic. If the two triangles are inscribed in a conic, their six sides touch another conic. Inverting and projecting:—if two groups of three conics each be passed through the three nodes of a quartic so that the three points of intersection of each group lie on the quartic, these six conics all touch another connodal trinodal quartic. A triangle is circumscribed about one conic, and two of its angular points are on a second conic; the locus of its third angular point is a conic.—Inverting and projecting:—if three conics be drawn through the three nodes of two connodal trinodal quartics so that they all touch one of the quartics and two of their points of intersection are on the other quartic, the locus of their third point of intersection is a connodal trinodal quartic. A triangle is inscribed in one conic and two of its sides touch a second conic; the envelope of its third side is a conic. Inverting and projecting:—if three conics be drawn through the three nodes of two connodal trinodal quartics so that their three points of intersection lie on one of the quartics and two of them touch the other quartic, the envelope of the third conic is another connodal trinodal quartic. The theorems of this section are stated in the most general terms and are still true when one or more of the nodes are changed into cusps. It is therefore not necessary to give separate theorems for the case of one cusp and two nodes. NODAL BICUSPIDAL QUARTICS. A quartic with one node and two cusps is a curve of the fourth class, having one double tangent and two points of inflection (see Salmon). Hence its reciprocal is also a nodal bicuspidal quartic, a fact of which frequent note will be made in this section. The inverse of a conic with respect to a focus is a curve called Pascal’s Limaçon. From the polar equation of a conic, the focus being the pole, it is evident that the polar equation of the limaçon may be written in the form: e 1 r = —— cos_x_ + —— ; p p where e and p are constants, being respectively the eccentricity and semi-latus rectum of the conic.