to the Cartesian Ovals in piano. 381 



parison with Dr. Salmon's ' Higher Plane Curves/ p. 175. 

 My focal cubic is the locus of one set of foci of a system of 

 conies whose axes are parallel, which pass therefore through 

 four points lying in a circle. The axis in which the foci are 

 taken, and which is parallel to the real asymptote, in general 

 meets the focal curve in two points. Whenever these points 

 come together, this parallel to the asymptote becomes a tan- 

 gent ; and the foci do come together for the circle itself and for 

 the three pairs of lines which can be drawn through the four 

 points in question. Hence the focal cubic not only passes through 

 the centre of the circle and through the intersections of the three 

 pairs of lines just spoken of, but at each of these four points is 

 parallel to the real asymptote, i. e. to the line bisecting one of the 

 angles in which the diagonals cross. It has also two circular 

 points at infinity. All these conditions are fulfilled by one of 

 Dr. Salmon's pair of circular cubics, of which the four points 

 in question are the foci. These curves are therefore identical ; 

 or, to express the same idea more fully, the two conjugate cir- 

 cular cubics, of which four points in a circle are the foci, together 

 constitute the complete locus of the foci of the system of conies 

 which can be drawn through those four points*. It is interesting, 

 moreover, to notice that the spherical curve which is the inter- 

 section of any two right cones with parallel axes, and which is 

 necessarily contained also in a third right cone fulfilling the same 

 condition, may be regarded as the inverse of any plane section of- 

 the spindle or Preformed by the revolution of a circle about an axis 

 cutting it, in respect to either point of intersection of the spindle 

 with its axis. The spherical curve in question is of course no 

 other than the so-called pair of twisted Cartesian ovals ; and its 

 focal curve may be any of Dr. Salmon's circular cubics of the 

 first kind, i. e. one whose four real foci lie in a circle. Finally, if a 

 double-curvature {i. e. twisted) Cartesian is given, we may define 

 its focal curve very simply as one of the two circular cubics of 

 which the points in which it is intersected by the plane passing 

 through the axes of its containing right cones are the four real 

 foci. The Cartesian itself is contained in a sphere, in a para- 

 boloid of revolution, in three right cones with parallel axesf, 



* Hence, as shown by Dr. Salmon, the focal cubic consists of an oval 

 and a serpentine branch. The two associated focal cubics, the same emi- 

 nent author has shown, may be regarded as the locus of the intersections of 

 similar conies having for their respective pairs of foci the two pairs of points 

 which make up the given set of four foci ; but their simpler geometrical 

 definition, as the complete locus of the foci of the conies drawn through 

 the four given points, appears to have escaped observation. 



t So remarkable is this property of the three cones, that, at the risk of 

 tedious reiteration, I think it desirable to present it under the same vivid 

 form in which it strikes my own mind. If any two indefinite straight lines 



