382 Prof. Sylvester on the Analogues in Space 



and also in three surfaces of revolution produced by the rotation 

 of cardioids (with their triple foci lying respectively at the points 

 of inflexion of the focal curve) about the stationary tangents*. 



If the two parabolas drawn through the four given points lying 

 in a circle which serve to determine the focal curve be 



x*+2ez + 2fy = 0, y <2 + 2gx + 2hy = 0, 



I find that the equation to the focal curve which is the locus 

 of the foci lying in the y axis of the conies drawn through the 

 four foci is 



h{x-e)(x* + y <2 ) + {fh-ke + kx)2-2(fh-ke + kx)hy-h?x* = 0. 



When x* + y 2 = 0, this gives {fh— ke + kx — hyf = 0, showing that 



x = 0, y = is a focus, which demonstrates the focal character of 

 each of the four fixed points. 



Departing from the theory of the quasi-Cartesian ovals, if in 

 general we take any four fixed points lying at the intersections 

 of the two conies, 



U = ax 1 -f by 1 + 2hxy + 2gzx + 2 fay, 

 and 



V = ax 1 + /fy 2 + 2f\xy + %*jzx + 2$xy, 



cross, they may be regarded as representing a couple of right cones gene- 

 rated respectively by the revolution of the lines about the two bisectors of 

 the angle which they form. Imagine now a quadrangle inscribed in a circle ; 

 its diagonals and pairs of opposite sides produced indefinitely will represent 

 three couples of right cones. This triad of couples may be resolved into a 

 couple of triads, the cones of each triad having their axes parallel inter se 

 and perpendicular to those of the other triad; the three cones of each triad 

 respectively will have a common intersection, the two intersections being 

 consociated twisted Cartesians whose focal cubics are respectively the two 

 consociated circular cubics of which the angles of the quadrangle are the 

 common foci. Moreover each such twisted Cartesian is a spherical curve 

 lying in the sphere of which the circle circumscribed about the quadrangle 

 is a great circle. The verification of these laws of intersection might be 

 used to form the subject of anew and instructive plate for students of 

 ordinary descriptive geometry. 



* It is interesting to trace the change of form in the general double- 

 curvature Cartesians in regard to the real and imaginary. For this purpose 

 conceive a sphere penetrated by a conical bodkin of indefinite length ; 

 when the point of the bodkin just pricks the sphere externally, the curve 

 consists of a single point ; as the bodkin is pushed in, the curve becomes 

 a single oval; when the point of the bodkin again meets the sphere inter- 

 nally, the curve will consist of an oval and a conjugate point; then two 

 ovals are formed ; then when the bodkin and sphere touch, of an oval 

 and a conjugate point ; then of a single oval ; and after the bodkin again 

 touches the sphere, in the other side, of a single point ; and finally the 

 curve returns wholly into the limbo of the imaginary, whence it originally 

 issued. There is apparently nothing analogous to this in the geometrical 

 genesis of the plane Cartesian ovals. 



