501 



-^= + -r^ +V^ — =0. (12) 



P — Po P - Po P — p.. 



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

 equations, may be supposed to be an ellipsoid, a hyperbo- 

 loid 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 Abe 

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

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

 cumscribing cone will remain unchanged, since the differences 

 of the quantities by which the squares of?, i\, Z are divided 

 are independent of the surface A. As p' is intermediate 

 in value between p and p", the normal to the hyperbo- 

 loid of one sheet is always the mean axis of the cone; the 

 focal lines lie in the plane ?^, and their equation is 



-r^ — +V^ = 0, (13) 



p' — p p — p" ^ ^ 



which shows that they are parallel to the asymptotes of a 

 central section made in the hyperboloid of one sheet by a 

 plane parallel to the plane KK, since the quantities p' — p and 

 p' — p" are (including the proper signs) the squares of the 

 semiaxes of the section which are parallel to ? and ^ re- 



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

 See Crelle's Journal, vol. xii., p. 137. See also the excellent work of M. 

 Chasles, published in 1837, and entitled "Aper9u historique sur I'Origine 

 etle Developpement des M^thodes en Geom^trie ;" 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 (Comptes rendus, torn, xvi., pp. 833, 1106); but that analogy had been 

 previously taught and developed in the lectures just alluded to. 



