On the Surfaces of the Second Order. 311 



is circumscribed by a cone having its vertex at S. If the nor- 

 mals applied at S to the given surfaces, taken in the order of the 

 equation (10), be the axes of new rectangular co-ordinates , ?j, , 

 the equation of the cone, referred to these co-ordinates, will be* 



V T? 2 3 



J^~Po + F^ + P^Po = 



The surfaces of the given system, in the order of their 

 equations, may be supposed to be an ellipsoid, a hyperboloid 

 of one sheet, and a hyperboloid of two sheets; the axes of 

 x, y, z being respectively the primary, the mean, and the 

 secondary axes of each surface. Then P is greater than P', 

 and P' greater than P". 



The normals to the given surfaces are the axes of the cone 

 expressed by the equation (12) ; and if the surface A be 

 changed, but still remain confocal with the given system, it 

 is obvious from that equation that the focal lines of the cir- 

 cumscribing cone will remain unchanged, since the differences 

 of the quantities by which the squares of , rj, are divided 

 are independent of the surface A. As P' is intermediate in 

 value between P and P", the normal to the hyperboloid of 

 one sheet is always the mean axis of the cone ; the focal lines 

 lie in the plane , and their equation is 



2 2 



p'_p + jy_ p" = 0> (13) 



* The equation (12) was obtained in the year 1832, and was given at my 

 Lectures in Hilary Term, 1836. The most remarkable properties of cones circum- 

 scribing confocal surfaces are immediate consequences of this equation. That 

 such cones, when they have a common vertex, are confocal, their focal lines 

 being the generatrices of the hyperboloid of one sheet passing through the 

 vertex, was first stated by Professor C. G. J. Jacobi, of Konigsberg, in 1834 

 (see Crelle's Journal, VOL. xn. p. 137). See also the excellent work of M. Chasles, 

 published in 1837, and entitled Aperqu historique sur VOriglne et U De'veloppe- 

 ment des Methodes en Geometrie, p. 387. The analogy which exists between the 

 focals of surfaces and the foci of curves of the second order was supposed by 

 M. Chasles to have been pointed out in that work for the first time (Qomptes 

 Rendus, torn, xvi., pp. 833, 1106) ; but that analogy had been previously taught and 

 developed in the Lectures just alluded to. 



