496 



in whatever direction the right line FF' passes through the 

 point N, it follows that the right line NS is an axis of the 

 cone which has the point S for its vertex and the focal for 

 its base. Further, if FF' intersect 6 in the point Q, we 

 have FN to F'N as FQ is to F'Q, because N is the pole of 

 9 with respect to the focal; therefore FQ is to F'Q as 

 FS is to F'S, and hence the right line QS also bisects one of 

 the angles made by FS and F'S. The right lines NS and 

 QS are therefore at right angles to each other, and as the 

 latter always lies in the tangent plane, the former must be 

 perpendicular to that plane. 



Consequently the normal at any point of a surface of the 

 second order is an axis of the cone which has that point for 

 its vertex and either of the focals for its base. 



It is known that when two confocal surfaces intersect 

 each other, they intersect everywhere at right angles ; and 

 that through any given point three surfaces may in general 

 be described, which shall have the same focal curves. If three 

 confocal surfaces pass through the point S, the normal to 

 each of them at S is an axis of each of the cones which stand 

 on the focals and have S for their common vertex. The 

 normals to the three surfaces are therefore the three axes of 

 each cone. 



If the points at which a series of confocal surfaces are 

 touched by parallel planes be the vertices of cones having 

 one of the focals for their common base, each of these cones 

 will have one of its axes perpendicular to the tangent planes. 

 Therefore when an axis of a cone which stands on a given 

 base is always parallel to a given right line, the locus of the 

 vertex is an equilateral hyperbola or a right line, according 

 as the base is a central conic or a parabola. 



§ 12. A system of three confocal surfaces intersecting 

 each other consists of an ellipsoid, a hyperboloid of one 

 sheet, and a hyperboloid of two sheets, if the focals be 

 central conies ; but it consists of two elliptic paraboloids 



