312 On the Surfaces of the Second Order. 



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 , 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- 

 spectively. The focal lines are therefore the generatrices of 

 that hyperboloid at the point S. 



When Bo = 0, the equation (12) becomes 



which is that of the cone standing on the focal ellipse and 

 having its vertex at S. When Q = 0, the same equation 

 becomes 



| + J+| = 0, (15) 



which is that of the cone standing on the focal hyperbola, 

 and having its vertex at S. The normal to the hyperboloid 

 of one sheet at the point S is the mean axis of both cones ; 

 the normal to the ellipsoid is the internal axis of the first 

 cone and the directive axis of the second, while the normal 

 to the hyperboloid of two sheets is the directive axis of the 

 first and the internal axis of the second. 



The three surfaces expressed by the equations 



_- 



P F F f Q Q' Q"""' 



(16) 



* + 5l + *L-l 

 M R' R" ~ 



are a confocal system, having their centres at S, and being re- 

 spectively an ellipsoid, a hyperboloid of one sheet, and a hyper- 

 boloid of two sheets. They intersect each other in the centre 

 of the system expressed by the equations (10), and their normals 

 at that point are the axes of #, /, z respectively. The relations 



