502 



spectively. The focal lines are therefore the generatrices 

 of that hyperboloid at the point S. 



When Rq :=. 0, the equation (12) becomes 



r + p + f^' = »' ('*) 



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

 having its vertex at S. When q© zz 0, the same equation 

 becomes 



- + -, + -^: = 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 nor- 

 mal 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 



^ . ^_ (16) 



are a confocal system, having their centre at S, and being 

 respectively an ellipsoid, a hyperboloid of one sheet, and a 

 hyperboloid 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 a;, y, z respectively. 

 The relations between the two systems of surfaces are there- 

 fore perfectly reciprocal. From the equations (14) and (15) 

 it is manifest that the asymptotic cones of the hyperboloids 

 of one system pass through the focals of the other. 



§ 16. The point S being the intersection of a given sys- 

 tem of confocal paraboloids whose equations are 



