Prof. Sylvester on Periodical Changes of Orbit, §c. [299 



come together, and the conic becomes a circle. Hence every 

 twisted Cartesian is the intersection of a sphere and a paraboloid 

 of revolution*. 



We are now in a position to turn back upon the focal cubic 

 itself and make it disclose its true nature ; for it will be no other 

 than one of the two curves of foci of the system of conies passing 

 through four points which lie in a circle. The axes of such a 

 system always retain their parallelism ; and consequently there 

 will be two separately determinable curves of foci — those, namely, 

 which lie in one set of parallel axes, and those which lie in the 

 other. By a general theorem of M. Chasles, the complete curve 

 of foci is of the sixth order, and consequently each of the two in 

 question ought to be, as we learn from the preceding theory it is, 

 a curve of only the third degree f. 



The equation of either may easily be found, and is of the form 



to which there is only one real asymptote, viz. x + D = 0. This, 

 then, is the general equation to the focal cubic to a twisted 

 Cartesian, and shows it to belong to the class of circular cubics. 



The focal cubic is or may be determined by a circle involving 

 three constants and four points arbitrarily chosen in the circle, 

 which, together with the three constants for fixing the plane of 

 the circle, give ten parameters in all. 



It passes through the intersections of the three pairs of oppo- 

 site sides of the quadrilateral inscribed in the circle, the centre of 

 the circle, and the two circular points at infinity ; the special re- 

 lations of the three intersections to the cubic await further inves- 

 tigation. The twisted cubic with which it is associated may be 

 determined by means of two right cones, each involving six con- 

 stants ; but as the axes must be coplanar and parallel, the num- 

 ber of parameters is reduced from twelve to ten, thus showing 

 that, when the focal curve is given, the associated ovals are deter- 

 mined (in this respect differing from the plane ovals, in which 

 one parameter remains indeterminate when the trifocal system 

 of points — the analogue of the focal cubic — is given) . It will 

 probably be found that when five points in the focal curve are 

 given, thus leaving two parameters disposable, the twisted ovals 

 drawn through any given point will cut each other orthogonally, 

 as Mr. Crofton has shown to be the case for the plane curves in 



* Or, as is evident from the text, the intersection of two (and therefore 

 also of three) right cones with parallel axes whose plane will contain the 

 focal cubic. 



t Every focal cubic to a given twisted Cartesian has thus its conjugate 

 corresponding to another twisted Cartesian, which may be regarded as the 

 conjugate of the first ; and the mutual relations of such curves seem to 

 invite further investigation. 



