298 Prof. Sylvester on Periodical Changes of Orbit, fyc. 



given twisted Cartesian possesses an infinite "number of foci, 

 every point that lies in a certain curve of the third degree being 

 a focus. When three foci are given there are four disposable 

 parameters, and no more, for determining this curve, which there- 

 fore cannot be any cubic curve, but is subject to satisfy two 

 conditions. This cubic curve of foci for the twisted Cartesian is 

 the analogue of the three focal points appertaining to the ordi- 

 nary plane Cartesian*. 



We are now in a position to obtain a much simpler mode of 

 genesis of the twisted Cartesian. If F, G, H be any three points 

 in a right line whose distances from each of a group of points in 

 a plane more than two in number are subject to two linear 

 relations, it is easy to prove that these latter will lie in a Car- 

 tesian oval, of which F, G, H are the three foci. If then we 

 draw any transversal in the plane of the focal cubic cutting it in 

 three points F, G, H, and make a plane revolve about this line, 

 each group of points in which the twisted curve is cut by this 

 revolving plane being subject to the same two linear conditions 

 of distance from F, G, H, they and therefore the entire twisted 

 curve will lie in a surface generated by the revolution of a certain 

 Cartesian oval about F, G, H. By drawing F, G, H parallel to 

 an asymptote f, one of the points, say H, goes off to infinity, and 

 F, G become the foci of a conic ; and as we may draw any other 

 transversal parallel to the former cutting the cubic in two other 

 points F', G', we learn that the twisted Cartesian is always expres- 

 sible as the intersection of two surfaces of revolution of the second 

 degree whose axes are parallel, and is thus a curve of only the 

 fourth order. It follows, moreover, that the focal cubic is the 

 locus of the foci of a family of conies in involution whose axes 

 are parallel. 



But we may still further simplify the conception of these 

 remarkable analogues to the ovals of Descartes. One of the 

 system of parallels last described will be the asymptote itself 

 meeting the cubic in only one point, so that the revolving conic 

 becomes a parabola ; and again, if we draw another transversal 

 parallel to the asymptote and touching the cubic, the two foci 



* It is due to Mr. Crofton to state that the idea which has led to the 

 discovery of this property of the twisted Cartesian was suggested by the 

 method employed by that excellent geometer for establishing the existence 

 of the third focus for the plane ovals, as described by him in a remarkable 

 paper on the theory of these curves read before the London Mathematical 

 Society on the 19th instant. It is important to notice that, since the dis- 

 tances of the points in the twisted curve from any one of the original foci 

 are linearly related to those from any other point L, and also from any 

 other point M in the focal cubic, the distances from L and M are themselves 

 linearly related. 



t It will presently appear that there is but one real asymptote to the 

 focal cubic. 



