3io On the Surfaces of the Second Order. 



one of them at any point S of the line of curvature formed by 

 their intersection lies in the tangent plane of the other, and is 

 parallel to an axis of any section made in the latter by a plane 

 parallel to the tangent plane. Supposing the surfaces to be cen- 

 tral, if two normals be applied at the point S, and a diameter of 

 each [surface be drawn parallel to the normal of the other, the 

 two diameters so drawn will be equal and of a constant length, 

 wherever the point S is taken on the line of curvature ; the 

 square of that length being equal to the difference of the squares 

 of the primary axes of the surfaces, and the diameter of the sur- 

 face which has the greater primary axis being real, while that of 

 the other surface is imaginary. As the point S moves along the 

 line of curvature, each constant diameter describes a cone condi- 

 rective with the surface to which it belongs ; the two cones so 

 described are reciprocal, and the focal lines of the cone which 

 belongs to one surface are perpendicular to the directive planes 

 of the other surface. 



When two confocal paraboloids intersect, if normals be ap- 

 plied to them at any point S of their intersection, and a bifocal 

 chord of each surface be drawn parallel to the normal of the 

 other, the two chords so drawn will be equal and of a constant 

 length, wherever the point S is taken in the line of intersection 

 of the surfaces ; that constant length being equal to the differ- 

 ence between the parameters of either pair of coincident principal 

 sections. 



15. The point S being the common intersection of a given 

 system of confocal surfaces, of which the equations are 



l ! 4. ?!. _ 1 f! + J^! 4. 1 _ i 



P Q K~ F Q' R'~ 



(10) 



suppose that another surface A confocal with these, and ex- 

 pressed by the equation 



<*"* tfl 9^ 



F+IT+JF-I' (11 > 



-to **o -"0 



